{ "what": "Lane S of the openai/math audit: every family with a Comparator challenge, its statements READ against the headline OpenAI wrote for it, the abstracts and OpenAI's scope note. A reading is not a certificate. Words, precedence and the second-reading rule are corpus/openai-math/preregistration.json (lanes.S, amendments 1 and 3); disagreements record the less severe word and show both.", "release": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb", "counts": { "families": 242, "decided": 242, "awaiting": 0, "words": { "MATCHES": 127, "NARROWER — UNLINKED": 39, "NARROWER — DECLARED": 38, "SUPPORT-ONLY": 4, "DIFFERENT": 11, "NARROWER — UNDECLARED": 23 }, "firstPassWords": { "MATCHES": 127, "NARROWER — UNDECLARED": 62, "NARROWER — DECLARED": 38, "SUPPORT-ONLY": 4, "DIFFERENT": 11 }, "secondReadings": 77, "flagReadings": 51, "flagDecisions": { "cleared": 76 }, "disagreements": 40, "disagreementsOnTheReading": [ "261" ], "conditionalOnLaneK": 9 }, "rows": [ { "family": "003", "title": "The quasi-Riemann hypothesis", "headline": "Proves that every Dirichlet L-function, including $`\\zeta(s)`$, is zero-free in $`\\Re s\\gt 7/8`$, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over $`\\mathbb Q(\\sqrt{-3})`$. A companion gives a different proof of the zero-free half-plane $`\\Re s\\gt 11/12`$.", "challenges": [ "QuasiRiemannHypothesis", "DirichletSevenEighths", "HeckeSevenEighths", "SiegelZeros" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that every Dirichlet L-function, including $`\\zeta(s)`$, is zero-free in $`\\Re s\\gt 7/8`$", "coveredBy": [ "DirichletSevenEighths", "QuasiRiemannHypothesis" ], "note": "Every modulus q >= 1 ([NeZero q]) and every character (primitive or not, principal included); only the principal pole chi = 1, s = 1 is excluded, which the abstract also allows (\"with the principal pole at s = 1 allowed\"). q = 1 already contains zeta; QuasiRiemannHypothesis states it separately." }, { "clause": "resolving the quasi-Riemann hypothesis", "coveredBy": [ "QuasiRiemannHypothesis", "DirichletSevenEighths" ], "note": "A fixed zero-free half-plane Re s > theta with theta = 7/8 < 1." }, { "clause": "The same half-plane is zero-free for every finite-order Hecke L-function over $`\\mathbb Q(\\sqrt{-3})`$", "coveredBy": [ "HeckeSevenEighths" ], "note": "Characters encoded as multiplicative characters of O/m trivial on the image of the units; K = Q(zeta_3) = Q(sqrt(-3)) has class number one, so these are exactly the finite-order Hecke characters. The L-function is an in-file theta/Mellin definition (see flag)." }, { "clause": "A companion gives a different proof of the zero-free half-plane $`\\Re s\\gt 11/12`$", "coveredBy": [ "DirichletSevenEighths", "QuasiRiemannHypothesis" ], "note": "As a statement, Re s > 11/12 is implied by the 7/8 statements; a \"different proof\" is not a statement. The 11/12 manuscript is not linked in the scope note, which is immaterial here." } ], "gap": null, "scopeNoteOnGap": "no gap. Context: docs/003.md:12 \"The principal-character poles at $s=1$ are excluded in the Dirichlet and Hecke statements.\"", "note": null, "firstSuspicious": "HeckeSevenEighths: the Hecke L-function is defined in the challenge, not imported. A second reader should verify the theta-lattice constants (lines 143-147, 194-202) against sum chi(a) N(a)^-s.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "HeckeSevenEighths", "flag": "DEFINITION (not a hole): the Hecke L-function is a bespoke in-file theta/Mellin construction, not a Mathlib object, and the challenge does not state its agreement with sum chi(a) N(a)^-s. My reading supports the lattice parametrization, but the constants (shift signs, the 1/2 factors, the /6) deserve a second reader." } ], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "005", "title": "Irrationality of Catalan’s constant", "headline": "Proves that Catalan's constant $`G=\\sum_{j\\ge0}(-1)^j/(2j+1)^2`$ is irrational.", "challenges": [ "Catalan" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that Catalan's constant $`G=\\sum_{j\\ge0}(-1)^j/(2j+1)^2`$ is irrational.", "coveredBy": [ "Catalan" ], "note": "Stated directly: Irrational of the real tsum over j : ℕ of (-1)^j/(2j+1)^2." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "Catalan", "flag": "JUNK? (not triggered): Mathlib's tsum is 0 on a non-summable family; here |(-1)^j/(2j+1)^2| ≤ 1/(2j+1)^2 is summable, so the tsum is the true value (and Irrational 0 would be false anyway)" } ], "flagReadings": [ { "challenge": "Catalan", "flag": "JUNK? (not triggered): Mathlib's tsum is 0 on a non-summable family; here |(-1)^j/(2j+1)^2| ≤ 1/(2j+1)^2 is summable, so the tsum is the true value (and Irrational 0 would be false anyway)", "decision": "cleared", "why": "The summand is (-1)^j / ((2j+1 : ℕ) : ℝ)^2, so its absolute value is 1/(2j+1)^2 ≤ 1/(j+1)^2: the family is absolutely summable and Mathlib's tsum is the genuine sum (Catalan's constant G). The junk-0 branch of tsum is never taken; had it been, Irrational 0 is false, so it could only have falsified the statement. Lane K has not run Catalan (notRunYet); this decision does not depend on it.", "evidence": [ "lean/ComparatorChallenges/Catalan.lean:8: Irrational (∑' j : ℕ, (-1 : ℝ) ^ j / ((2 * j + 1 : ℕ) : ℝ) ^ 2) := by" ] } ], "readings": [ "first/batch1.json", "flags/flags1.json" ] }, { "family": "007", "title": "Ordinary two-point correlations and the corrected Elliott conjecture", "headline": "Proves the ordinary two-point Chowla conjecture, with a bound $`O(X/(\\log X)^c)`$ for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times $`n^{it}`$ for $`|t|\\le X`$.", "challenges": [ "OrdinaryElliott", "OrdinaryTwoPointCorrelations" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves the ordinary two-point Chowla conjecture, with a bound $`O(X/(\\log X)^c)`$ for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute.", "coveredBy": [ "OrdinaryTwoPointCorrelations" ], "note": "liouville_log_saving: ∃ c > 0, ∀ nonproportional forms (a₁n+b₁, a₂n+b₂), ∃ C > 0, ∀ X ≥ 3, |Σ_{n≤⌊X⌋} λ(a₁n+b₁)λ(a₂n+b₂)| ≤ C·X/(log X)^c. Quantifier order matches the prose: c is chosen before the forms (absolute), C after them. The two-point Chowla case is a₁=a₂=1, b₁=0, b₂=h." }, { "clause": "More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times $`n^{it}`$ for $`|t|\\le X`$.", "coveredBy": [ "OrdinaryElliott", "OrdinaryTwoPointCorrelations" ], "note": "Both files state: one-bounded multiplicative f₁, f₂, at least one uniformly nonpretentious (∀ q > 0, ∀ χ mod q, inf over |t| ≤ N of D(f, χ·n^{it}; N) → ∞), distinct shifts ⇒ (1/N)Σ_{n≤N} f₁(n+h₁)f₂(n+h₂) → 0; OrdinaryTwoPointCorrelations adds the nonproportional-affine-form version." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "OrdinaryElliott", "flag": "JUNK? checked, none: sInf is over the image of the nonempty compact interval [−N, N] of a nonnegative function (each summand 1 − Re(...) ≥ 0 because |f(p)| ≤ 1), so it is a genuine infimum; Real.sqrt is of a nonnegative quantity." }, { "challenge": "OrdinaryTwoPointCorrelations", "flag": "JUNK? checked, none: division by (N : ℂ) is junk only at N = 0, irrelevant to the limit; Real.rpow (Real.log X) c with X ≥ 3 has a positive base." } ], "flagReadings": [ { "challenge": "OrdinaryElliott", "flag": "JUNK? checked, none: sInf is over the image of the nonempty compact interval [−N, N] of a nonnegative function (each summand 1 − Re(...) ≥ 0 because |f(p)| ≤ 1), so it is a genuine infimum; Real.sqrt is of a nonnegative quantity.", "decision": "cleared", "why": "Each summand (1 − Re(f p · conj(χ(p) p^{it})))/p is ≥ 0, because ‖f p‖ ≤ 1 (OneBounded), |χ(p)| ≤ 1 and |p^{it}| = 1. distanceSq is therefore ≥ 0 and Real.sqrt is genuine. The sInf is taken over the image of the nonempty compact interval [−N, N] (it contains 0) under a function that is continuous in t and bounded below by 0. It is the genuine infimum, and in fact a minimum. When χ(p) = 0 the summand is 1/p, which is the standard convention. A junk sInf would have sat in the hypothesis UniformlyNonpretentious, but none occurs.", "evidence": [ "lean/ComparatorChallenges/OrdinaryElliott.lean:11: def OneBounded (f : ℕ → ℂ) : Prop := ∀ n, ‖f n‖ ≤ 1", "lean/ComparatorChallenges/OrdinaryElliott.lean:19: (1 - (f p * star (χ (p : ZMod q) *", "lean/ComparatorChallenges/OrdinaryElliott.lean:20: Complex.exp ((t * Real.log (p : ℝ) : ℝ) * Complex.I))).re) / (p : ℝ)", "lean/ComparatorChallenges/OrdinaryElliott.lean:23: (t X : ℝ) : ℝ := Real.sqrt (distanceSq f χ t X)", "lean/ComparatorChallenges/OrdinaryElliott.lean:27: Tendsto (fun N : ℕ => sInf ((fun t : ℝ => distance f χ t (N : ℝ)) ''", "lean/ComparatorChallenges/OrdinaryElliott.lean:28: Set.Icc (-(N : ℝ)) (N : ℝ))) atTop atTop" ] }, { "challenge": "OrdinaryTwoPointCorrelations", "flag": "JUNK? checked, none: division by (N : ℂ) is junk only at N = 0, irrelevant to the limit; Real.rpow (Real.log X) c with X ≥ 3 has a positive base.", "decision": "cleared", "why": "Division by (N : ℂ) is junk only at N = 0, which is one term of a sequence whose limit is taken at atTop. In Real.rpow (Real.log X) c with X ≥ 3, the base is log X ≥ log 3 > 1 > 0, so (log X)^c is a genuine positive power and the bound C·X/(log X)^c is genuine. f, g and liouville are evaluated only at a·n + b ≥ 1 or n + h ≥ 1 (a ≥ 1, n ≥ 1), so their junk value at 0 is never used. characterTwist's log is evaluated only at primes inside squaredDistance.", "evidence": [ "lean/ComparatorChallenges/OrdinaryTwoPointCorrelations.lean:14: ∑ n ∈ Finset.Icc 1 N, f (n + h₁) * g (n + h₂)", "lean/ComparatorChallenges/OrdinaryTwoPointCorrelations.lean:17: ∑ n ∈ Finset.Icc 1 N, f (a₁ * n + b₁) * g (a₂ * n + b₂)", "lean/ComparatorChallenges/OrdinaryTwoPointCorrelations.lean:46: ∃ C : ℝ, 0 < C ∧ ∀ X : ℝ, 3 ≤ X →", "lean/ComparatorChallenges/OrdinaryTwoPointCorrelations.lean:47: ‖affineSum liouville liouville a₁ a₂ b₁ b₂ ⌊X⌋₊‖ ≤", "lean/ComparatorChallenges/OrdinaryTwoPointCorrelations.lean:48: C * X / Real.rpow (Real.log X) c := by", "lean/ComparatorChallenges/OrdinaryTwoPointCorrelations.lean:56: Tendsto (fun N : ℕ => correlationSum f₁ f₂ h₁ h₂ N / (N : ℂ)) atTop (nhds 0) := by" ] } ], "readings": [ "first/batch5.json", "flags/flags2.json" ] }, { "family": "008", "title": "The Deligne–Drinfeld conjecture", "headline": "Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture.", "challenges": [ "DeligneDrinfeld" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "is freely generated by one element in each odd weight 3, 5, 7", "coveredBy": [ "DeligneDrinfeld" ], "note": "W = solutions in Lie_Q(x,y) of antisymmetry, the three-term relation and the four-strand pentagon in T4 (Drinfeld's grt_1 in graded form); e is a graded linear iso from the free Lie algebra on N (generator k in weight 2k+3) onto W carrying brackets to the Ihara bracket." }, { "clause": "with the Ihara bracket", "coveredBy": [ "DeligneDrinfeld" ], "note": "ihara p q = D p q - D q p + [p,q] with D_p(x) = 0, D_p(y) = [y,p] (tangentLift)." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "009", "title": "Function-field reconstruction from Milnor K-theory and Galois data", "headline": "Reconstructs function fields of transcendence degree at least two over algebraically closed constants from $`K^{\\mathrm M}_1/\\ell`$, $`K^{\\mathrm M}_2/\\ell`$, and their product. These data recover the perfect closure and constants when ℓ differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-ℓ Galois data away from the characteristic.", "challenges": [ "BogomolovPopInjectivity" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Reconstructs function fields of transcendence degree at least two over algebraically closed constants from $`K^{\\mathrm M}_1/\\ell`$, $`K^{\\mathrm M}_2/\\ell`$, and their product.", "status": "unlinked", "coveredBy": [], "manuscript": "Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026", "evidence": [ "corpus/openai-math/release.json:5496: \"dir\": \"Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026\"", "corpus/openai-math/release.json:5498: together with their full bilinear product, determine the perfect closure and its constant field.", "lean/docs/009.md:3: The following describes the scope of the Lean formalization related to the following accompanying paper(s):", "lean/docs/009.md:5: - [The Bogomolov-Pop reconstruction theorem](../../preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf)", "lean/ComparatorChallenges/BogomolovPopInjectivity.lean: no Milnor K-group appears; the only challenge of the family is Galois-theoretic (lines 117-124)" ] }, { "clause": "These data recover the perfect closure and constants when ℓ differs from the characteristic,", "status": "unlinked", "coveredBy": [], "manuscript": "Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026", "evidence": [ "corpus/openai-math/release.json:5498: over arbitrary algebraically closed fields of characteristic different from ℓ.", "lean/docs/009.md:5: - [The Bogomolov-Pop reconstruction theorem](../../preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf)" ] }, { "clause": "and the original field and its named base in equal characteristic.", "status": "unlinked", "coveredBy": [], "manuscript": "Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026", "evidence": [ "corpus/openai-math/release.json:5507: \"dir\": \"Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026\"", "corpus/openai-math/release.json:5509: We prove that every isomorphism of their first Milnor K-groups modulo p preserving the degree-two Steinberg relations is a nonzero scalar multiple of the map induced by a unique field isomorphism.", "lean/docs/009.md:5: - [The Bogomolov-Pop reconstruction theorem](../../preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf)" ] }, { "clause": "Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-ℓ Galois data away from the characteristic.", "status": "declared", "coveredBy": [ "BogomolovPopInjectivity" ], "manuscript": "The-Bogomolov-Pop-reconstruction-theorem-September-23-2026", "evidence": [ "lean/docs/009.md:11: The selected statement proves this uniqueness; existence of a field isomorphism for every admissible Galois-data isomorphism is outside it.", "lean/ComparatorChallenges/BogomolovPopInjectivity.lean:123: ∀ (a a' : IsomF k K l L) (b : IsomCUnits ell K L),", "lean/ComparatorChallenges/BogomolovPopInjectivity.lean:124: PhiGraph a b → PhiGraph a' b → a = a' := by" ] } ], "gap": "Both Milnor K-theory reconstruction theorems (mod ℓ with ℓ ≠ char; mod p in equal characteristic) are not stated at all. Of Bogomolov–Pop only injectivity is stated (existence omitted, declared).", "scopeNoteOnGap": "Milnor K clauses: silent. Bogomolov–Pop existence: lean/docs/009.md:11: The selected statement proves this uniqueness; existence of a field isomorphism for every admissible Galois-data isomorphism is outside it.", "note": "Only the injectivity half of Bogomolov–Pop is stated (existence declared out at 009.md:11); both Milnor K-theory clauses come from the two October-5 manuscripts the scope note does not link, so the most severe status is UNLINKED. I differ from the first reader only because amendment 3 split UNLINKED from UNDECLARED; on the facts we agree.", "firstSuspicious": "Pattern seen across this batch: the CONTENTS.md headline was widened to manuscripts the scope note does not link, and the scope note never says those manuscripts are unformalized. Strictly that is silence, hence UNDECLARED; a second reader should weigh whether the scope note's paper list counts as a declaration. Here: the two October 5 Milnor K-theory papers.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json", "second/second1.json" ] }, { "family": "012", "title": "Independent largest prime factors of consecutive integers", "headline": "Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and $`n+1`$ are asymptotically independent in ordinary natural density. In particular, the integers satisfying $`P^+(n)\\lt P^+(n+1)`$ have density 1/2.", "challenges": [ "JointDickman" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and $`n+1`$ are asymptotically independent in ordinary natural density.", "coveredBy": [ "JointDickman" ], "note": "joint_law: for every 0 PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) :", "lean/ComparatorChallenges/DukePrimeDegree.lean:208: ∃ ν : Measure (X (d+1)), IsHaarProbability ν ∧", "lean/ComparatorChallenges/DukePrimeDegree.lean:210: IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by", "lean/ComparatorChallenges/DukePrimeDegree.lean:165: {O | ∃ M' : FullLattice (d + 1) K, LocallyHomothetic M M' ∧" ], "note": "Encoding: X = SL_n(Z)\\SL_n(R) via rightRel; Haar = right-invariant probability; packet = every lattice locally homothetic to M at every prime, with all sign translates, orbit-volume weighted; 'no escape of mass' = IsTightMeasureSet; fields vary along the sequence (K : ℕ → Type u); the headline's gloss 'the relevant order ... discriminant tends to infinity' is hD (|disc K|·[O_K : multiplier order]²). Junk defaults (orbitalHaar = 0 without a fundamental domain) cannot satisfy the conclusion, which asserts every packet measure is a probability measure." }, { "clause": "arbitrary-order Picard packets in primitive quartic fields", "status": "unlinked", "coveredBy": [], "manuscript": "Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026", "evidence": [ "lean/docs/015.md:5: - [Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type](../../preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf)", "lean/ComparatorChallenges/DukePrimeDegree.lean:201: (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)", "CONTENTS.md:429: Let K range over totally real quartic fields with no proper intermediate field, and let $`\\mathcal O`$ be any order in K." ], "note": "Degree 4 is excluded by hprime/hfive; the quartic manuscript is not among the papers docs/015.md links." }, { "clause": "maximal-order ideal-class packets in primitive sextic fields", "status": "unlinked", "coveredBy": [], "manuscript": "Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026", "evidence": [ "lean/docs/015.md:5: - [Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type]", "CONTENTS.md:438: We prove that the complete ideal-class packets of maximal orders in totally real sextic fields with no proper intermediate fields become equidistributed, with no escape of mass" ], "note": "Degree 6 is not prime, so excluded by hprime; the sextic manuscript is not linked." } ], "gap": "The primitive quartic (arbitrary orders) and primitive sextic (maximal orders) equidistribution theorems are not stated; only prime degree ≥ 5.", "scopeNoteOnGap": "silent — the note links and describes only the prime-degree paper: lean/docs/015.md:9: The formalization proves equidistribution of volume-weighted torus packets for totally real number fields of every fixed prime degree at least five.", "note": "The prime-degree clause is stated in full; the quartic and sextic clauses come from the two October-5 manuscripts that docs/015.md does not link, so the gap is UNLINKED. The first reader found the same gap and called it UNDECLARED only because amendment 3's UNLINKED word did not exist yet.", "firstSuspicious": "Pattern seen across this batch: the CONTENTS.md headline was widened to manuscripts the scope note does not link, and the scope note never says those manuscripts are unformalized. Strictly that is silence, hence UNDECLARED; a second reader should weigh whether the scope note's paper list counts as a declaration. Here: the two October 5 quartic and sextic papers.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json", "second/second2.json" ] }, { "family": "017", "title": "The irrationality exponent of π is 2", "headline": "Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational $`p/q`$ satisfies $`|\\pi-p/q|\\ge q^{-2-\\varepsilon}`$. This also proves convergence of the Flint–Hills series $`\\sum_{n\\ge1}1/(n^3\\sin^2 n)`$, with angles in radians.", "challenges": [ "PiExponent" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational $`p/q`$ satisfies $`|\\pi-p/q|\\ge q^{-2-\\varepsilon}`$.", "coveredBy": [ "PiExponent" ], "note": "First conjunct: ∀ ν > 2 ∃ Q ≥ 2 ∀ p q, Q ≤ q → q^(−ν) ≤ |π − p/q| (ν = 2 + ε; quantifier order ∀ε ∃Q matches 'for every ε … all sufficiently large q'). Second conjunct pins the exponent at exactly 2 via sSup of the exponents with infinitely many approximations." }, { "clause": "This also proves convergence of the Flint–Hills series $`\\sum_{n\\ge1}1/(n^3\\sin^2 n)`$, with angles in radians.", "coveredBy": [], "note": "Not stated by the challenge; the scope note declares it out (lean/docs/017.md:11)." } ], "gap": "Convergence of the Flint–Hills series Σ 1/(n³ sin² n) is claimed in the headline but not stated by PiExponent.", "scopeNoteOnGap": "The paper's convergence consequence for the Flint–Hills series is outside this selected statement.", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "PiExponent", "flag": "JUNK? (checked, benign): sSup over ℝ is 0 for an unbounded set, but conjunct (i) bounds the set above by 2 and Dirichlet puts (0,2) inside it, so sSup is the genuine supremum; a junk 0 would make the claim false, not weaker." } ], "flagReadings": [ { "challenge": "PiExponent", "flag": "JUNK? (checked, benign): sSup over ℝ is 0 for an unbounded set, but conjunct (i) bounds the set above by 2 and Dirichlet puts (0,2) inside it, so sSup is the genuine supremum; a junk 0 would make the claim false, not weaker.", "decision": "cleared", "why": "Real sSup is junk (0) only for an empty or unbounded-above set; this set is neither. Nonempty and containing (0,2): for 0<ν<2 the continued-fraction convergents p/q of π with q≥2 (infinitely many, reduced so r.den = q, and |π−r|>0 since π is irrational) satisfy |π−p/q| < 1/q² < q^(−ν). Bounded above by 2: an element ν>2 would need infinitely many reduced r with |π−r| < den(r)^(−ν) ≤ 1; only finitely many rationals of bounded denominator lie within distance 1 of π, so den(r)→∞ along them, and once den(r) ≥ Q this contradicts the theorem's own conjunct (i) at that ν (lines 8-10, which quantify over ALL integer pairs with q ≥ Q, reduced ones included). Independently of (i), π's irrationality measure is already known to be finite, so the set is bounded whatever the truth of (i). sSup is therefore the genuine supremum, and a junk 0 ≠ 2 could only falsify the claim. q ≥ Q ≥ 2 and den ≥ 2 also keep every rpow base positive. Lane K: CERTIFIED.", "evidence": [ "ComparatorChallenges/PiExponent.lean:8: (∀ ν : ℝ, 2 < ν → ∃ Q : ℤ, 2 ≤ Q ∧", "ComparatorChallenges/PiExponent.lean:9: ∀ p q : ℤ, Q ≤ q →", "ComparatorChallenges/PiExponent.lean:10: (q : ℝ) ^ (-ν) ≤ |Real.pi - (p : ℝ) / (q : ℝ)|) ∧", "ComparatorChallenges/PiExponent.lean:11: sSup {ν : ℝ | 0 < ν ∧", "ComparatorChallenges/PiExponent.lean:12: Set.Infinite {r : ℚ | 2 ≤ r.den ∧", "ComparatorChallenges/PiExponent.lean:13: 0 < |Real.pi - (r : ℝ)| ∧", "ComparatorChallenges/PiExponent.lean:14: |Real.pi - (r : ℝ)| < (r.den : ℝ) ^ (-ν)}} = 2 := by" ] } ], "readings": [ "first/batch3.json", "flags/flags3.json" ] }, { "family": "020", "title": "Squarefree quartics and power-free polynomial values", "headline": "Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the $`(d-2)`$-power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4.", "challenges": [ "PowerFreeValues" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density.", "coveredBy": [ "PowerFreeValues" ], "note": "Case natDegree = 4 of allDegrees (k = d-2 = 2)." }, { "clause": "More generally, establishes the $`(d-2)`$-power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4.", "coveredBy": [ "PowerFreeValues" ], "note": "allDegrees states every d ≥ 4 directly (it does not stop at 8 or appeal to Browning), so it states at least the clause." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "PowerFreeValues", "flag": "JUNK? (not triggered): Mathlib's tprod of a non-multipliable family is a junk value; the statement asserts Multipliable and 0 < densityConstant explicitly" } ], "flagReadings": [ { "challenge": "PowerFreeValues", "flag": "JUNK? (not triggered): Mathlib's tprod of a non-multipliable family is a junk value; the statement asserts Multipliable and 0 < densityConstant explicitly", "decision": "cleared", "why": "The junk branch of tprod (Mathlib's tprod_eq_one_of_not_multipliable: the value is 1 for a non-multipliable family, which would make '0 < densityConstant' trivially true) is shut by the statement itself: DensityStatement's first conjunct asserts Multipliable (localFactor f k), so densityConstant is the true limit of the finite partial products; its second conjunct 0 < densityConstant then excludes the product-converges-to-0 case. Every factor is genuinely positive under the hypothesis LocallyAdmissible (rho f (p^k) < p^k). Both are conclusions, so a non-convergent product could only make the theorem false, never vacuous; lane K: CERTIFIED.", "evidence": [ "ComparatorChallenges/PowerFreeValues.lean:28-29: noncomputable def densityConstant (f : Polynomial ℤ) (k : ℕ) : ℝ := ⏎ ∏' p : Nat.Primes, localFactor f k p", "ComparatorChallenges/PowerFreeValues.lean:31-33: def DensityStatement (f : Polynomial ℤ) (k : ℕ) : Prop := ⏎ Multipliable (localFactor f k) ∧ ⏎ 0 < densityConstant f k ∧", "ComparatorChallenges/PowerFreeValues.lean:22-23: def LocallyAdmissible (f : Polynomial ℤ) (k : ℕ) : Prop := ⏎ ∀ p : ℕ, p.Prime → rho f (p ^ k) < p ^ k", "ComparatorChallenges/PowerFreeValues.lean:25-26: noncomputable def localFactor (f : Polynomial ℤ) (k : ℕ) (p : Nat.Primes) : ℝ := ⏎ 1 - (rho f (p.val ^ k) : ℝ) / (p.val : ℝ) ^ k" ] } ], "readings": [ "first/batch1.json", "flags/flags4.json" ] }, { "family": "021", "title": "A quadratic bound for Jacobsthal’s function", "headline": "Answers Jacobsthal's quadratic-bound question: every interval of $`Ck^2`$ consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss.", "challenges": [ "Jacobsthal", "JacobsthalImproved" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "every interval of $`Ck^2`$ consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C", "coveredBy": [ "Jacobsthal", "JacobsthalImproved" ], "note": "Jacobsthal states exactly ∃ C>0 ∀ k≥1 ∃ m ≤ C·k² with every window {a,…,a+m−1} (a ∈ ℤ) containing an integer coprime to every n>0 with ≤ k distinct prime factors. JacobsthalImproved states the paper's stronger Ck²/(log log 3k)² bound (which implies the quadratic one since (log log 3k)² is bounded below by (log log 3)² > 0)." }, { "clause": "The bound is uniform over prime sets and interval positions", "coveredBy": [ "Jacobsthal", "JacobsthalImproved" ], "note": "C is quantified before k, n and a (∃ C ∀ k ∀ n ∀ a): uniform." }, { "clause": "removes the classical logarithmic loss", "coveredBy": [ "Jacobsthal", "JacobsthalImproved" ], "note": "The quadratic bound (and its iterated-log improvement) has no log k factor." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "023", "title": "Patterson's first moment for cubic Gauss sums", "headline": "Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order $`X^{5/6}/\\log X`$. Every fixed nonzero prime-angle Fourier mode has smaller order.", "challenges": [ "PattersonFirstMoment" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order $`X^{5/6}/\\log X`$", "coveredBy": [ "PattersonFirstMoment" ], "note": "Main term (6/5)·c*·X^{5/6}/log X with c* = (2π)^{2/3}/(3Γ(2/3)), error o(X^{5/6}/log X); sum over every p with 3 | p − 1 and Prime p in Z[ω] (both conjugates)." }, { "clause": "Every fixed nonzero prime-angle Fourier mode has smaller order.", "coveredBy": [ "PattersonFirstMoment" ], "note": "patterson_angularCancellation: for each ℓ ≠ 0, Σ (p/|p|)^ℓ G(p) = o(X^{5/6}/log X)." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "024", "title": "An asymptotic formula for the number of totients", "headline": "Gives an asymptotic equivalent for the number $`V(x)`$ of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, $`V(cx)/V(x)\\to c`$ for every fixed c > 0, answering Erdős and Hall’s scaling question.", "challenges": [ "TotientAsymptotic", "TotientCompanionZero" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Gives an asymptotic equivalent for the number $`V(x)`$ of distinct totient values up to x", "coveredBy": [ "TotientAsymptotic" ], "note": "Third conjunct of totient_asymptotic_formula: V x / mainTerm x -> 1, where V x counts the totient values in [1, floor x]." }, { "clause": "with a positive bounded phase-dependent factor determined by convergent arithmetic approximations", "coveredBy": [ "TotientAsymptotic" ], "note": "mainTerm x = x/log x * G(x, m x) * A(1)(theta x). The first conjunct gives uniform convergence of the finite approximants AH H to A on [0,1); the second gives cMinus > 0 with cMinus <= A s <= cPlus for all s in [0,1)." }, { "clause": "In particular, $`V(cx)/V(x)\\to c`$ for every fixed c > 0, answering Erdős and Hall’s scaling question", "coveredBy": [ "TotientAsymptotic" ], "note": "Fourth conjunct: for all c > 0, V(c x)/V(x) -> c." } ], "gap": null, "scopeNoteOnGap": "no gap (docs/024.md:9 describes exactly this; the companion results of :11 are extras)", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "025", "title": "Short Egyptian fractions", "headline": "Every rational $`a/b`$ with $`1\\le a\\lt b`$ is a sum of $`O(\\log\\log b)`$ distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions.", "challenges": [ "EgyptianFractions", "ShortEgyptianFractions" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Every rational $`a/b`$ with $`1\\le a\\lt b`$ is a sum of $`O(\\log\\log b)`$ distinct positive unit fractions.", "coveredBy": [ "ShortEgyptianFractions" ], "note": "main conjunct 1 gives an expansion for every 1<=a= b0) bounds every minLength a b, and minLength is attained because the set is nonempty by conjunct 1. Denominators >= 2 and strictly increasing = distinct positive unit fractions other than 1/1, which cannot occur when a/b<1." }, { "clause": "The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions.", "coveredBy": [ "ShortEgyptianFractions", "EgyptianFractions" ], "note": "lower bound c1*log log b <= maxMinLength b for b >= b0; EgyptianFractions.main_double_log_order restates both bounds with Fin-indexed expansions." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "ShortEgyptianFractions", "flag": "JUNK? cleared: minLength is sInf of a subset of ℕ (sInf ∅ = 0), but conjunct 1 proves nonemptiness for every 1<=a minLength a b)", "lean/ComparatorChallenges/ShortEgyptianFractions.lean:18: (∀ a b : ℕ, 1 ≤ a → a < b → ∃ ns : List ℕ, IsExpansion a b ns) ∧" ] }, { "challenge": "EgyptianFractions", "flag": "JUNK? cleared: F k = ncard would be 0 for an infinite set, but one_expansions_finite_and_bounded proves finiteness; v k = sInf of a set proved nonempty in missing_denominator_semantics; Filter.liminf/limsup on ℝ are meaningful only for bounded sequences, and prescribed_denominator_corollary's own first and third conjuncts bound log log v(k)/k eventually, so the liminf claims are not junk.", "decision": "cleared", "why": "F k = ncard, which is 0 on infinite sets, but one_expansions_finite_and_bounded (65) proves Set.Finite (OneExpansions k) for every k. v k = sInf, which is 0 on ∅, but missing_denominator_semantics (101) proves the set nonempty. On ℝ, liminf and limsup are junk only for a sequence that is not eventually bounded. The first conjunct of prescribed_denominator_corollary (109-110) gives exp(exp(k/600)) ≤ v k ≤ 1 + k^(2^(k−1)) eventually, so 1/600 ≤ prescribedSlope k ≤ ((k−1)·log 2 + log log k + o(1))/k. The sequence is therefore eventually bounded, and the liminf/limsup at 111-114 are the true ones. Separately, an unbounded sequence would give the junk 0 and falsify log 2/257 ≤ liminf, so the junk could not weaken anyway. Lane K has not run EgyptianFractions.", "evidence": [ "lean/ComparatorChallenges/EgyptianFractions.lean:33: (OneExpansions k).ncard", "lean/ComparatorChallenges/EgyptianFractions.lean:65: (∀ k : ℕ, Set.Finite (OneExpansions k) ∧", "lean/ComparatorChallenges/EgyptianFractions.lean:101: (∃ m : ℕ, m ∈ missingDenominators k) ∧", "lean/ComparatorChallenges/EgyptianFractions.lean:109: Real.exp (Real.exp ((k : ℝ) / 600)) ≤ (v k : ℝ) ∧", "lean/ComparatorChallenges/EgyptianFractions.lean:110: v k ≤ 1 + k ^ (2 ^ (k - 1))) ∧", "lean/ComparatorChallenges/EgyptianFractions.lean:111: (Real.log 2 / 257 ≤ Filter.liminf prescribedSlope Filter.atTop ∧" ] } ], "readings": [ "first/batch8.json", "flags/flags1.json" ] }, { "family": "026", "title": "Positive lower density of large prime gaps", "headline": "For every fixed C > 0, a positive proportion of consecutive prime gaps exceed $`C\\log p_n`$, throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where $`p_n/n`$ increases have positive lower density, answering Erdős and Prachar.", "challenges": [ "PrimeGaps" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "For every fixed C > 0, a positive proportion of consecutive prime gaps exceed $`C\\log p_n`$, throughout every sufficiently large initial segment of the primes. The proportion may depend on C.", "coveredBy": [ "PrimeGaps" ], "note": "First conjunct: ∀ C > 0, ∃ c > 0, ∃ N₀, ∀ N ≥ N₀, c·N ≤ #{n ∈ [1,N] : C·log p_n < p_{n+1} − p_n}. Quantifier order ∀C ∃c matches 'the proportion may depend on C'; 'every sufficiently large initial segment' = ∀ N ≥ N₀; strict '>' matches 'exceed'." }, { "clause": "Consequently, the indices where $`p_n/n`$ increases have positive lower density, answering Erdős and Prachar.", "coveredBy": [ "PrimeGaps" ], "note": "Second conjunct: 0 < lowerAsymptoticDensity {n ≥ 1 | p_n/n < p_{n+1}/(n+1)}." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "PrimeGaps", "flag": "JUNK checked, none: the sSup is over a set containing 0 and bounded above by 1 (count ≤ N for N ≥ 1), so it is the genuine lim inf and '0 <' it means some positive d is an eventual lower bound; N = 0 division is avoidable by N₀ ≥ 1" }, { "challenge": "PrimeGaps", "flag": "NOTE: the scope note describes only the ratio corollary (026.md:9), but the challenge also states the large-gap clause — an understatement by the note, not a gap" } ], "flagReadings": null, "readings": [ "first/batch6.json" ] }, { "family": "027", "title": "Potential integral density on curve character varieties", "headline": "Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SLr character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.", "challenges": [ "CharacterVarietiesAllSeamsSupport" ], "word": "SUPPORT-ONLY", "firstWord": "SUPPORT-ONLY", "secondWord": "SUPPORT-ONLY", "disagreement": null, "clauses": [ { "clause": "Resolves the determinant-one curve case of Litt's integral-density question.", "status": "declared", "coveredBy": [], "manuscript": "Integral-points-on-character-varieties-of-curves-September-25-2026", "evidence": [ "lean/docs/027.md:11: This is conditional compatibility of the produced solution. Potential integral density, including the prescribed-boundary and all-component conclusions, remains outside the selected statement.", "lean/ComparatorChallenges/README.md:14: These setups verify supporting results, not the corresponding papers’ main", "lean/ComparatorChallenges/README.md:17: - `CharacterVarietiesAllSeamsSupport.json`: compatibility of the produced marked" ] }, { "clause": "For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SLr character variety over the full ring of integers of one number field.", "status": "declared", "coveredBy": [], "manuscript": "Integral-points-on-character-varieties-of-curves-September-25-2026", "evidence": [ "lean/docs/027.md:11: Potential integral density, including the prescribed-boundary and all-component conclusions, remains outside the selected statement." ] }, { "clause": "Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.", "status": "declared", "coveredBy": [], "manuscript": "Integral-points-on-character-varieties-of-curves-September-25-2026", "evidence": [ "lean/docs/027.md:11: including the prescribed-boundary and all-component conclusions, remains outside the selected statement." ] } ], "gap": "the entire headline: potential Zariski density of integral points on SL_r character varieties is not stated; the challenge proves only that the produced marked values satisfy every seam equation of the produced marked diagram", "scopeNoteOnGap": "This is conditional compatibility of the produced solution. Potential integral density, including the prescribed-boundary and all-component conclusions, remains outside the selected statement. (lean/docs/027.md:11); also lean/ComparatorChallenges/README.md:14-18 (line breaks joined): \"These setups verify supporting results, not the corresponding papers’ main theorems: - `CharacterVarietiesAllSeamsSupport.json`: compatibility of the produced marked solution with every seam equation.\"", "note": "The only challenge (CharacterVarietiesAllSeamsSupport: the produced marked values satisfy every seam equation) is on the README's supporting-result list, and the scope note names every headline clause as outside the statement; SUPPORT-ONLY, agreeing with the first reader. Flag STATEMENT IMPORTS THE SOLUTION LIBRARY: the challenge's line 1 imports OAI.AlgebraicGeometry.CharacterVarieties.Seams.ProducedEquations, so its definitions are the claimant's own (lean/ComparatorChallenges/CharacterVarietiesAllSeamsSupport.lean:1).", "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "CharacterVarietiesAllSeamsSupport", "flag": "STATEMENT IMPORTS SOLUTION-SIDE LIBRARY: the challenge file imports OAI.AlgebraicGeometry.CharacterVarieties.Seams.ProducedEquations, so ProducedMarkedSeamEquations and everything it reaches (producedMarkedDiagram, producedMarkedValues, SameFramedFlag, ...) is defined in OpenAI's own library tree, not in the challenge; the statement's meaning is fixed by those library definitions (lean/OAI/AlgebraicGeometry/CharacterVarieties/Seams/ProducedEquations.lean:25-30), which a reader must also audit" }, { "challenge": "CharacterVarietiesAllSeamsSupport", "flag": "SUPPORT: declared a supporting result in lean/ComparatorChallenges/README.md" } ], "flagReadings": null, "readings": [ "first/batch8.json", "second/second3.json" ] }, { "family": "028", "title": "Uniformly bounded components of Gaussian-prime graphs", "headline": "Proves the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have uniformly bounded steps. More strongly, for every distance bound D, the graph joining Gaussian primes at distance at most D has uniformly bounded finite component sizes, depending only on D, including primes on the coordinate axes.", "challenges": [ "GaussianMoat" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "no infinite walk through distinct Gaussian primes can have uniformly bounded steps", "coveredBy": [ "GaussianMoat" ], "note": "MainEndpoint: no injective sequence of irreducible Gaussian integers with all steps ≤ D, for every real D." }, { "clause": "for every distance bound D, the graph joining Gaussian primes at distance at most D has uniformly bounded finite component sizes, depending only on D, including primes on the coordinate axes", "coveredBy": [ "GaussianMoat" ], "note": "UniformEndpoint: ∀ D ∃ B, every component is finite with ncard ≤ B and every injective D-step walk has length ≤ B." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "033", "title": "Iitaka subadditivity, variation, and logarithmic additivity", "headline": "Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class-$`\\mathcal C`$ manifolds with rational simple-normal-crossing boundaries. For projective fibrations $`f:U\\to V`$ of smooth complex quasi-projective varieties with connected fibers, general fiber F, and $`\\bar\\kappa(V)\\ge0`$, proves Popa's inequality $`\\bar\\kappa(U)\\ge\\kappa(F)+\\max\\{\\bar\\kappa(V),\\mathop{\\mathrm{Var}}\\nolimits (f)\\}`$, where variation measures the whole geometric generic fiber.", "challenges": [ "LogKodairaFiberNegative" ], "word": "DIFFERENT", "firstWord": "DIFFERENT", "secondWord": "DIFFERENT", "disagreement": null, "clauses": [ { "clause": "Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class-$`\\mathcal C`$ manifolds with rational simple-normal-crossing boundaries.", "status": "unlinked", "coveredBy": [], "manuscript": "Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026", "evidence": [ "CONTENTS.md:816: We prove Campana's orbifold Iitaka subadditivity conjecture for rational simple normal crossing boundaries on compact manifolds in Fujiki class $`\\mathcal C`$, including coefficient one.", "lean/docs/033.md:5: - [The reverse logarithmic Kodaira inequality and additivity](../../preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026/paper.pdf)", "lean/ComparatorChallenges/LogKodairaFiberNegative.lean:237: ∀ F : FiberModel f.toBoundaryFibration y, F.boundary.kodaira = ⊥ →" ] }, { "clause": "For projective fibrations $`f:U\\to V`$ of smooth complex quasi-projective varieties with connected fibers, general fiber F, and $`\\bar\\kappa(V)\\ge0`$, proves Popa's inequality $`\\bar\\kappa(U)\\ge\\kappa(F)+\\max\\{\\bar\\kappa(V),\\mathop{\\mathrm{Var}}\\nolimits (f)\\}`$, where variation measures the whole geometric generic fiber.", "status": "unlinked", "coveredBy": [], "manuscript": "Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026", "evidence": [ "CONTENTS.md:825: We prove the logarithmic Iitaka–Viehweg inequality for projective surjective morphisms with connected fibers between smooth complex quasi-projective varieties whose base has nonnegative logarithmic Kodaira dimension.", "lean/docs/033.md:5: - [The reverse logarithmic Kodaira inequality and additivity](../../preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026/paper.pdf)", "lean/ComparatorChallenges/LogKodairaFiberNegative.lean:238: E.kodaira = D.kodaira + F.boundary.kodaira ∧" ] } ], "gap": "Neither headline sentence is stated. The family's only challenge states the negative-fibre branch of logarithmic Kodaira ADDITIVITY (for a very general complex base point y outside the base boundary, if the fibre pair has log Kodaira dimension ⊥ = −∞ then κ̄(X,E) = κ̄(Y,D) + κ̄(F) = −∞ and every positive-degree log pluriform on X vanishes), for stratum-smooth surjective connected-fibre morphisms of smooth projective reduced-SNC pairs — a result of the third manuscript ('The reverse logarithmic Kodaira inequality and additivity'), which the headline does not mention (only the family title's word 'logarithmic additivity' does). It neither implies nor is implied by the headline: subadditivity is a lower bound on κ̄(U) and says nothing when κ(F) = −∞, while the Lean statement is an upper bound (equality to −∞).", "scopeNoteOnGap": "lean/docs/033.md:11: \"The finite-dimension and negative-base branches of the paper's additivity theorem are outside this selected statement.\" — the note links only the reverse-logarithmic-Kodaira manuscript (033.md:5) and is silent on Campana's conjecture and Popa's inequality.", "note": "Neither headline sentence is stated, and both come from manuscripts the scope note does not link. The single challenge states the linked third paper's negative-fibre branch of log Kodaira additivity (for very general y off D: κ̄(F)=−∞ ⇒ κ̄(X,E)=κ̄(Y,D)+κ̄(F)=−∞ and every positive-degree log pluriform vanishes), an upper-bound/vanishing statement that neither implies nor is implied by the two lower-bound (subadditivity) clauses, so DIFFERENT by the pre-registered definition. Agree with the first reader; under amendment 3 both clauses are UNLINKED, so the DIFFERENT records that the headline describes papers the formalization never claimed, not a silence of the note about its linked paper.", "firstSuspicious": "Word sits on the DIFFERENT / SUPPORT boundary: the challenge is not listed in ComparatorChallenges/README.md's supporting-result list, and the scope note calls it a 'selected statement', not a supporting one. If the family TITLE ('…and logarithmic additivity') is read as part of the headline, the additivity part becomes a declared partial cover.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "LogKodairaFiberNegative", "flag": "VACUITY? (low): the per-point claim quantifies ∀ F : FiberModel f y; at a point with no fibre model it would hold vacuously — for a very general point of a stratum-smooth fibration a smooth projective fibre with induced reduced SNC boundary exists, so the content is not lost" }, { "challenge": "LogKodairaFiberNegative", "flag": "QUANTIFIER: VeryGenerally Y (fun y => …) = ∃ countable proper closed Z, ∀ complex y outside — matches the manuscript's 'very general base point'; the hypothesis κ(F_y) = ⊥ is per-point, so one very general fibre with κ = −∞ forces κ̄(X,E) = −∞" } ], "flagReadings": null, "readings": [ "first/batch6.json", "second/second4.json" ] }, { "family": "039", "title": "Nagata’s conjecture and maximal Seshadri constants", "headline": "Proves Nagata's strict inequality $`\\sum_i m_i\\lt d\\sqrt r`$ for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. It also proves maximal multipoint Seshadri constants $`(L^n/r)^{1/n}`$ for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ, and at the geometric generic tuple over any algebraically closed field of positive characteristic.", "challenges": [ "Nagata", "MaximalSeshadriConstants" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves Nagata's strict inequality $`\\sum_i m_i\\lt d\\sqrt r`$ for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi.", "status": "stated", "coveredBy": [ "Nagata" ], "reading": "nagata_conjecture: for every count ≥ 10, one ℕ-indexed family of proper Zariski-closed sets of ordered distinct configurations in (P²_ℂ)^count with a common non-member, chosen before every curve and multiplicity vector; for configurations outside it, every effective cycle of homogeneous prime principal ideals (reducible, non-reduced included) and every m with m_i ≤ mult satisfies Σ m_i < deg·√count. nagata_conjecture_coordinates gives the homogeneous-polynomial form. Encoding: very general = outside a countable union of proper closed sets; degree = totalDegree of an equation chosen by Classical.choose (unique up to scalar).", "evidence": [ "lean/ComparatorChallenges/Nagata.lean:352: theorem nagata_conjecture :", "lean/ComparatorChallenges/Nagata.lean:359: ∀ curve : Workers.W10.EffectivePlaneCurve, ∀ multiplicities : Fin count → ℕ,", "lean/ComparatorChallenges/Nagata.lean:361: (∑ index, (multiplicities index : ℝ)) <", "lean/ComparatorChallenges/Nagata.lean:362: (Workers.W10.curveDegree curve : ℝ) * Real.sqrt (count : ℝ) := by" ] }, { "clause": "It also proves maximal multipoint Seshadri constants $`(L^n/r)^{1/n}`$ for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ", "part": "n = 2 (surfaces) over ℂ", "status": "stated", "coveredBy": [ "MaximalSeshadriConstants" ], "reading": "maximalSeshadriConstants: for every smooth integral surface closed-immersed in complex projective space and every ample L, ∃ r₀ > 0 ∀ r ≥ r₀ a countable family of proper closed subsets of distinct r-tuples with a common non-member, outside which the multipoint Seshadri constant equals √(L²/r) (and the boundary class on the blowup is nef). Encoding: L² and L·C as Euler-characteristic combinations (finrank of Ext groups); multiplicity as ideal order in the local ring.", "evidence": [ "lean/ComparatorChallenges/MaximalSeshadriConstants.lean:42: smooth : SmoothOfRelativeDimension 2 structureMap", "lean/ComparatorChallenges/MaximalSeshadriConstants.lean:401: seshadriConstant S L r p = Real.sqrt ((selfIntersection S L : ℝ) / r)", "lean/ComparatorChallenges/MaximalSeshadriConstants.lean:408: theorem maximalSeshadriConstants (S : Surface) (L : LineBundle S.scheme)" ] }, { "clause": "It also proves maximal multipoint Seshadri constants $`(L^n/r)^{1/n}`$ for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ", "part": "n ≥ 3 over ℂ", "status": "unlinked", "coveredBy": [], "manuscript": "Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions-October-5-2026", "evidence": [ "lean/ComparatorChallenges/MaximalSeshadriConstants.lean:38: structure Surface where", "lean/ComparatorChallenges/MaximalSeshadriConstants.lean:42: smooth : SmoothOfRelativeDimension 2 structureMap", "CONTENTS.md:1126: [Maximal Multipoint Seshadri Constants in Higher Dimensions](preprints/Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions-October-5-2026/main.pdf)", "CONTENTS.md:1128: Let L be an ample line bundle on a smooth integral complex projective variety X of dimension n ≥ 3.", "lean/docs/039.md:5: - [Nagata's conjecture for plane curves](../../preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026/main.pdf)", "lean/docs/039.md:6: - [Maximal Seshadri constants on arbitrary polarized surfaces](../../preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026/main.pdf)", "lean/docs/039.md:14: The formalized result establishes this for every smooth integral complex projective surface $S$ and ample line bundle $L$." ] }, { "clause": "and at the geometric generic tuple over any algebraically closed field of positive characteristic.", "status": "unlinked", "coveredBy": [], "manuscript": "Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026", "evidence": [ "lean/ComparatorChallenges/MaximalSeshadriConstants.lean:40: structureMap : scheme ⟶ Spec (CommRingCat.of ℂ)", "lean/ComparatorChallenges/MaximalSeshadriConstants.lean:19: Proj (MvPolynomial.homogeneousSubmodule (Fin (N + 1)) ℂ)", "CONTENTS.md:1135: [Maximal multipoint Seshadri constants in positive characteristic](preprints/Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026/seshadri-positive-characteristic.pdf)", "CONTENTS.md:1137: the ordinary multipoint Seshadri constant at the geometric generic tuple of r points equals the volume bound $`(L^n/r)^{1/n}`$. The same conclusion holds in dimension two.", "lean/docs/039.md:5: - [Nagata's conjecture for plane curves](../../preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026/main.pdf)", "lean/docs/039.md:6: - [Maximal Seshadri constants on arbitrary polarized surfaces](../../preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026/main.pdf)" ] } ], "gap": "maximal multipoint Seshadri constants in dimension n >= 3 (complex, very general points) and in positive characteristic (all n >= 2, geometric generic tuple) are claimed by the headline but not stated; Lean states only smooth complex projective surfaces", "scopeNoteOnGap": "silent — lean/docs/039.md links only the two September-23 manuscripts (Nagata; surfaces) and restricts positively: \"The formalized result establishes this for every smooth integral complex projective surface $S$ and ample line bundle $L$.\" (docs/039.md:14); it never mentions dimension n >= 3, positive characteristic, or the two October-5 manuscripts (Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions, Maximal-multipoint-Seshadri-constants-in-positive-characteristic) that the headline now draws on", "note": "Nagata and the complex-surface Seshadri result are stated. Dimension n ≥ 3 over ℂ and every positive-characteristic case (n ≥ 2, geometric generic tuple) are not stated; they come from the two October-5 manuscripts, which docs/039.md does not link (it links only the two September-23 papers), so the gap is UNLINKED rather than UNDECLARED. The first reader recorded UNDECLARED before amendment 3 introduced UNLINKED; on the same facts the word is now NARROWER — UNLINKED.", "firstSuspicious": "DECLARED vs UNDECLARED is a judgement call: docs/039.md:14 restricts the formalized result to surfaces, which a careful reader can compare against the headline, but the scope note never names the n >= 3 or positive-characteristic results as omitted and does not link their two October-5 manuscripts (the scope note appears to predate the headline's expansion). A second reader should decide whether a positive restriction counts as 'says what is left out'.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "Nagata", "flag": "VACUITY? cleared: 'exceptional index ≠ Set.univ' plus an explicit common non-member rule out the trivial all-exceptional family" }, { "challenge": "MaximalSeshadriConstants", "flag": "JUNK? (low): every intersection number is a Module.finrank of an Abelian.Ext group with a ℂ-module structure obtained by Module.compHom through Γ(X,⊤); finrank is 0 for an infinite-dimensional module, so the encoding is only as faithful as the identification Ext^n(O,M) = H^n(X,M) with finite-dimensional values for coherent sheaves; I did not find a degeneracy, but this is a heavy bespoke encoding of L^2 and L·C" }, { "challenge": "MaximalSeshadriConstants", "flag": "QUANTIFIER: ∃ r0 ∀ r >= r0 ∃ exceptional family — matches 'all sufficiently large r'; the threshold depends on (S,L), as the scope note says" } ], "flagReadings": null, "readings": [ "first/batch8.json", "second/second5.json" ] }, { "family": "047", "title": "Zariski cancellation and affine fibrations over the complex numbers", "headline": "Constructs an integral complex affine fourfold $`X\\not\\cong\\mathbb A^4`$ with $`X\\times\\mathbb A^1\\cong\\mathbb A^5`$, disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections $`X\\to\\mathbb A^1`$ and $`\\mathbb A^5\\to\\mathbb A^2`$ have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.", "challenges": [ "ComplexCancellation" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Constructs an integral complex affine fourfold $`X\\not\\cong\\mathbb A^4`$ with $`X\\times\\mathbb A^1\\cong\\mathbb A^5`$, disproving affine-space cancellation over ℂ in dimension four.", "status": "stated", "coveredBy": [ "ComplexCancellation" ], "encoding": "X = Spec A encoded by its coordinate ring A = ℂ[p,s,u,F,J]/(H); X×A¹ ≅ A⁵ as Polynomial A ≃ₐ[ℂ] MvPolynomial (Fin 5) ℂ; X ≇ A⁴ as no ℂ-algebra isomorphism with MvPolynomial (Fin 4) ℂ; integral fourfold = finite type, domain, ringKrullDim 4", "evidence": [ "lean/ComparatorChallenges/ComplexCancellation.lean:20-23: def MainStatement : Prop := ⏎ Algebra.FiniteType ℂ A ∧ IsDomain A ∧ ringKrullDim A = 4 ∧ ⏎ Nonempty (Polynomial A ≃ₐ[ℂ] MvPolynomial (Fin 5) ℂ) ∧ ⏎ ¬ Nonempty (A ≃ₐ[ℂ] MvPolynomial (Fin 4) ℂ)" ] }, { "clause": "It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections $`X\\to\\mathbb A^1`$ and $`\\mathbb A^5\\to\\mathbb A^2`$ have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.", "status": "undeclared", "coveredBy": [], "manuscript": "An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026", "evidence": [ "preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/build/main.tex:13-17: The same construction disproves the Stable Coordinate Conjecture in ⏎ ambient dimension five and yields smooth $\\mathbb A^3$-fibrations over ⏎ $\\mathbb A^1$ and $\\mathbb A^2$ that are not Zariski-locally trivial. ⏎ These fibrations disprove the Dolgachev--Weisfeiler affine-fibration ⏎ conjecture over these bases.", "preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/build/sections/07-affine-fibrations.tex:21-28: Both are smooth surjective morphisms of relative dimension three. ⏎ For either morphism and every point $y$ of its base, the fiber over $y$ ⏎ is isomorphic to $\\Aff^3_{\\kappa(y)}$; in particular, every geometric ⏎ fiber is affine three-space. Neither morphism is a ⏎ Zariski-locally trivial $\\Aff^3$-bundle. ⏎ Thus the Dolgachev--Weisfeiler affine-fibration conjecture fails in ⏎ relative dimension three over each of the smooth complex affine bases ⏎ $\\Aff^1$ and $\\Aff^2$.", "lean/docs/047.md:5: - [An explicit failure of complex affine-space cancellation](../../preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/paper.pdf)", "lean/docs/047.md:9: Affine-space cancellation asks whether $X\\times\\mathbb A^1\\cong\\mathbb A^{n+1}$ forces $X\\cong\\mathbb A^n$. The formalized result gives an explicit finite-type complex domain $A$ of Krull dimension four with $A[w]\\cong\\mathbb C[x_1,\\ldots,x_5]$ but $A\\not\\cong\\mathbb C[x_1,\\ldots,x_4]$. Thus adjoining one variable erases a genuine algebraic distinction.", "lean/docs/047.md:11: The separate stable-coordinate and general line-bundle lifting consequences are not included.", "preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/build/sections/04-bundle.tex:193-194: on $Y$ would not suffice. Instead, we prove a lifting statement for any ⏎ line bundle over a smooth integral affine base. The two ingredients", "lean/ComparatorChallenges/ComplexCancellation.lean:20-23: def MainStatement : Prop := ⏎ Algebra.FiniteType ℂ A ∧ IsDomain A ∧ ringKrullDim A = 4 ∧ ⏎ Nonempty (Polynomial A ≃ₐ[ℂ] MvPolynomial (Fin 5) ℂ) ∧ ⏎ ¬ Nonempty (A ≃ₐ[ℂ] MvPolynomial (Fin 4) ℂ)" ] } ], "gap": "The Dolgachev–Weisfeiler clause (smooth surjections X → 𝔸¹ and 𝔸⁵ → 𝔸² with all residue-field fibers 𝔸³ that are not Zariski-locally trivial) is not stated.", "scopeNoteOnGap": "silent", "note": "The cancellation counterexample is stated exactly (as ℂ-algebras). The Dolgachev–Weisfeiler corollary (07-affine-fibrations.tex, Corollary \"Nontrivial smooth A³-fibrations\") is in the one linked paper and in no statement; the note's exclusion names only \"stable-coordinate and general line-bundle lifting consequences\", and the general line-bundle lifting is the paper's linearization proposition (04-bundle.tex:193-194), a proof device, not the fibration corollary — the note's title alone (\"... and affine fibrations ...\") does not declare. Agree with the first reader.", "firstSuspicious": "The scope note's title is 'Zariski cancellation and affine fibrations over the complex numbers' (lean/docs/047.md:1), yet its body never mentions fibrations and the only Comparator statement is the cancellation counterexample. A second reader should confirm that 'general line-bundle lifting consequences' (047.md:11) does not refer to the fibration corollary; in the manuscript it does not (07-affine-fibrations.tex vs 01-introduction.tex:189).", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "ComplexCancellation", "flag": "NARROWER — UNDECLARED: the second headline sentence (Dolgachev–Weisfeiler fibrations) has no Lean counterpart in this or any family-047 challenge, and lean/docs/047.md does not mention fibrations." } ], "flagReadings": null, "readings": [ "first/batch3.json", "second/second6.json" ] }, { "family": "049", "title": "A stable-coordinate counterexample in four variables", "headline": "Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine three-space, yet none of its embeddings is rectifiable, also disproving the Abhyankar–Sathaye conjecture even when all fibers are affine spaces.", "challenges": [ "AbhyankarSathaye", "CommutingDerivations", "StableCoordinateFour" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables.", "coveredBy": [ "StableCoordinateFour" ], "note": "PolynomialPackage: f explicit, totalDegree 5, ∃ ℂ-algebra automorphism φ of ℂ[x₀..x₄] with φ(f) = X 0, and no automorphism of ℂ[x₀..x₃] sends f to X 0." }, { "clause": "Every fiber is affine three-space, yet none of its embeddings is rectifiable,", "coveredBy": [ "StableCoordinateFour" ], "note": "FiberPackage: ∀ t : ℂ, ℂ[x]/(f − t) ≃ₐ ℂ[y₁,y₂,y₃] and no automorphism maps the ideal (f − t) onto (X 0)." }, { "clause": "also disproving the Abhyankar–Sathaye conjecture even when all fibers are affine spaces.", "coveredBy": [ "StableCoordinateFour", "AbhyankarSathaye" ], "note": "Follows inside StableCoordinateFour (fibre t = 0 is A³ but f is not a coordinate); AbhyankarSathaye states the noncoordinate-with-affine-quotient polynomial for every n ≥ 4." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch6.json" ] }, { "family": "050", "title": "A counterexample to Griffiths’ positivity conjecture", "headline": "Constructs ample rank-two bundles on $`\\mathbb P^1\\times\\mathbb P^1`$ with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.", "challenges": [ "QuadricBundles" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs ample rank-two bundles on $`\\mathbb P^1\\times\\mathbb P^1`$", "coveredBy": [ "QuadricBundles" ], "note": "∃ G and a sequence E m of AlgebraicRankTwoBundle (cocycle data of 2×2 regular transition matrices over principal opens of OnePoint ℂ × OnePoint ℂ) with E m the power pullback of G twisted by O(1,1), and IsAmple (E m) for every m>0. IsAmple = some positive symmetric power has finitely many global sections giving a base-point-free, injective, closed-image, locally-invertible map — very ampleness of O(n) on the projectivization, which is equivalent to ampleness." }, { "clause": "with no smooth Hermitian metric of strictly Griffiths-positive curvature", "coveredBy": [ "QuadricBundles" ], "note": "∃ m₀>0 ∀ m ≥ m₀, ¬∃ h : SmoothHermitianMetric (E m) with h.StrictlyGriffithsPositive. The metric is a C^∞ positive-definite matrix field on the dual bundle per chart, compatible on overlaps; strict Griffiths positivity is PosDef of ∂_u∂̄_u K − (∂_u K)^* K^{-1} ∂_u K for every u ≠ 0 at every point. I re-derived that this is −⟨Θξ,ξ⟩ of the dual metric K, so PosDef ⇔ V* Griffiths-negative ⇔ V Griffiths-positive: the sign convention is right." }, { "clause": "disproving Griffiths' positivity conjecture already on the quadric surface", "coveredBy": [ "QuadricBundles" ], "note": "Follows from the two clauses above (a single ample bundle with no such metric is enough)." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": "The whole conclusion rests on hand-built definitions (AlgebraicRankTwoBundle, IsAmple via GivesClosedProjectiveEmbedding, SmoothHermitianMetric, StrictlyGriffithsPositive). I checked the curvature sign/order and that the metric structure is inhabited for any honest cocycle (partition of unity), so 'no metric' is not vacuous; a second reader should confirm GivesClosedProjectiveEmbedding cannot be met by a non-ample bundle (it is the clause that, if too weak, would make the counterexample easier).", "confidence": "medium", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "052", "title": "Tangent splittings and product decompositions", "headline": "A splitting of the tangent bundle of a compact Kähler manifold into two integrable holomorphic subbundles induces a compatible product decomposition of its universal cover, proving the two-summand form of Beauville's splitting conjecture. On smooth rationally connected projective manifolds, both summands are automatically integrable, establishing Höring's conjecture and the corresponding product decomposition.", "challenges": [ "KahlerSplitting", "SplitTangentIntegrability" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "A splitting of the tangent bundle of a compact Kähler manifold into two integrable holomorphic subbundles induces a compatible product decomposition of its universal cover", "coveredBy": [ "KahlerSplitting" ], "note": "Compact connected X (connectedness as in the abstract), a Kahler metric as data, positive ranks r1 and r2, and both summands integrable. The universal cover Z is biholomorphic to Y1 x Y2 (connected and simply connected), and the differential carries the lifted summands to the factor directions." }, { "clause": "proving the two-summand form of Beauville's splitting conjecture", "coveredBy": [ "KahlerSplitting" ], "note": "Same theorem." }, { "clause": "On smooth rationally connected projective manifolds, both summands are automatically integrable, establishing Höring's conjecture", "coveredBy": [ "SplitTangentIntegrability" ], "note": "Compact connected complex manifold, dim >= 2, a projective embedding witnessing rational connectedness, any holomorphic tangent splitting with positive ranks: both summands are integrable." }, { "clause": "and the corresponding product decomposition", "coveredBy": [], "note": "Not stated. The union does not yield it either. KahlerSplitting needs a KahlerMetric (Fubini-Study pullback, not stated), and the two challenges use different structures with no bridge between them: UniversalCoverSplitting.ComplexManifold vs SplitTangent.ComplexManifold, chart-wise Integrable vs mlieBracketWithin Integrable, HolomorphicSplitting vs TangentSplitting." } ], "gap": "The product decomposition for rationally connected projective manifolds (the last clause) is not stated; neither are the Kahler paper's further corollaries.", "scopeNoteOnGap": "docs/052.md:14: \"The paper's compatible product decomposition is a separate consequence, outside this selected statement.\" (and docs/052.md:12: \"Automatic integrability and the paper's additional corollaries are not included.\" for the Kahler paper)", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "058", "title": "Semialgebraic universal covers and bounded domains", "headline": "Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products $`D\\times\\mathbb C^m\\times F`$, with D bounded symmetric and F simply connected, normal, and projective. A universal cover is quasi-projective exactly when the bounded symmetric factor is absent. In particular, a smooth projective variety covered by ℂn has a finite étale cover by an abelian variety.", "challenges": [ "SymmetricDomains" ], "word": "DIFFERENT", "firstWord": "DIFFERENT", "secondWord": "DIFFERENT", "disagreement": null, "clauses": [ { "clause": "Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products $`D\\times\\mathbb C^m\\times F`$, with D bounded symmetric and F simply connected, normal, and projective.", "status": "unlinked", "coveredBy": [], "manuscript": "Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026", "evidence": [ "CONTENTS.md:1522: [Semialgebraic universal covers of normal projective varieties](preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/paper.pdf)", "CONTENTS.md:1524: We prove the Kollár–Pardon conjecture: the universal cover of a connected normal projective complex variety is biholomorphic to a semialgebraic open subset of a projective variety if and only if it is a product of a bounded symmetric domain, a complex affine space, and a simply connected normal projective variety.", "preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/build/main.tex:92: \\begin{theorem}[The Koll\\'ar--Pardon classification]\\label{thm:main}", "preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/build/main.tex:102: \\widetilde X\\simeq D\\times\\C^m\\times F.", "lean/docs/058.md:3: The following describes the scope of the Lean formalization related to the following accompanying paper(s):", "lean/docs/058.md:5: - [Symmetry of semialgebraic bounded domains with compact quotient](../../preprints/Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026/paper.pdf)", "lean/docs/058.md:9: The formalized result answers the paper's symmetry question affirmatively.", "lean/ComparatorChallenges/SymmetricDomains.lean:47: theorem main {n : ℕ} (V U : Set (Affine n))", "lean/ComparatorChallenges/SymmetricDomains.lean:56: IsSmooth U ∧ ∃ (m : ℕ) (D : Set (Affine m)),", "lean/ComparatorChallenges/SymmetricDomains.lean:57: IsBoundedSymmetricDomain D ∧ Nonempty (Biholomorph U D) := by" ], "note": "No universal cover, projective variety, affine factor or product decomposition occurs in the challenge." }, { "clause": "A universal cover is quasi-projective exactly when the bounded symmetric factor is absent.", "status": "unlinked", "coveredBy": [], "manuscript": "Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026", "evidence": [ "preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/build/sections/quasiprojective.tex:10: \\begin{corollary}[Quasi-projective covers]\\label{cor:quasiprojective}", "preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/build/sections/quasiprojective.tex:14: \\item \\(U\\) is biholomorphic to a quasi-projective variety.", "preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/build/sections/quasiprojective.tex:18: \\item For some integer \\(m\\geq0\\) and some simply connected projective", "preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/build/sections/quasiprojective.tex:21: U\\simeq\\C^m\\times F.", "lean/docs/058.md:5: - [Symmetry of semialgebraic bounded domains with compact quotient](../../preprints/Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026/paper.pdf)", "lean/docs/058.md:9: The formalized result answers the paper's symmetry question affirmatively." ], "note": "The quasi-projectivity criterion is the KP manuscript's Corollary cor:quasiprojective; not stated." }, { "clause": "In particular, a smooth projective variety covered by ℂn has a finite étale cover by an abelian variety.", "status": "unlinked", "coveredBy": [], "manuscript": "Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026", "evidence": [ "preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/build/sections/quasiprojective.tex:54: \\begin{corollary}[Projective affine-space uniformization]", "preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/build/sections/quasiprojective.tex:57: \\(n\\geq0\\). If its ordinary universal cover is biholomorphic to \\(\\C^n\\),", "preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/build/sections/quasiprojective.tex:58: then there is a finite \\'etale Galois cover \\(A\\to X\\) with \\(A\\) an", "lean/docs/058.md:5: - [Symmetry of semialgebraic bounded domains with compact quotient](../../preprints/Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026/paper.pdf)", "lean/docs/058.md:9: The formalized result answers the paper's symmetry question affirmatively." ], "note": "The KP manuscript's Corollary cor:affine-uniformization; not stated." } ], "gap": "All three headline clauses (the classification, the quasi-projectivity criterion, the C^n corollary). The single challenge states the companion paper's theorem instead: a connected bounded semialgebraic domain, relatively open in an affine algebraic set and carrying a proper cocompact holomorphic discrete action, is smooth and biholomorphic to a bounded symmetric domain. It neither implies the classification nor follows from it as stated. U need not be simply connected, the action need not be free, and the ambient set may be singular, so U is not presented as the universal cover of a normal projective variety.", "scopeNoteOnGap": "silent. docs/058.md links only the symmetry paper (line 5) and describes only it (line 9: \"The formalized result answers the paper's symmetry question affirmatively.\"). It never mentions the Kollar-Pardon classification, the quasi-projectivity criterion or the C^n corollary.", "note": "No headline clause is stated, and the one challenge states the linked symmetry paper's Theorem 1.1 (a bounded semialgebraic domain, relatively open in a possibly singular affine algebraic set, with a proper cocompact holomorphic discrete action, is smooth and biholomorphic to a bounded symmetric domain), which neither implies the Kollár–Pardon classification nor follows from it as stated (U need not be simply connected, the action need not be free, the ambient set may be singular; the KP manuscript derives the symmetry of its bounded factor inside its own argument, main.tex:213-214) — so DIFFERENT by the precedence rule. Every headline clause comes from the KP manuscript, which docs/058.md does not link: the formalization never claimed the headline; the DIFFERENT word records that the headline OpenAI wrote and its only statement are about different theorems, not an undeclared gap. Agree with the first reader.", "firstSuspicious": "DIFFERENT vs NARROWER — UNDECLARED: I chose DIFFERENT because the bounded-domain theorem is not a special case of the headline as stated (non-simply-connected U, non-free action, singular ambient). A reader who treats it as a component of the headline's proof might write NARROWER — UNDECLARED. Either way, none of the headline clauses is stated and the scope note is silent on them.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "SymmetricDomains", "flag": "DIFFERENT: the challenge states the companion bounded-domain theorem, not the headline classification. See the gap and the evidence from both sides." } ], "flagReadings": null, "readings": [ "first/batch7.json", "second/second7.json" ] }, { "family": "066", "title": "Bounded klt complements for Fano contractions", "headline": "Proves the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture for ϵ-lc complex Fano-type pairs with nef anti-log-canonical divisor. For ϵ-lc Fano contractions over any algebraically closed characteristic-zero field, it gives klt complements near every base point, with index bounded only by dimension and positive rational ϵ.", "challenges": [ "CartierChartCompactnessSupport" ], "word": "SUPPORT-ONLY", "firstWord": "SUPPORT-ONLY", "secondWord": "SUPPORT-ONLY", "disagreement": null, "clauses": [ { "clause": "Proves the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture for ϵ-lc complex Fano-type pairs with nef anti-log-canonical divisor.", "status": "declared", "coveredBy": [], "manuscript": "Bounded-klt-complements-for-Fano-contractions-September-25-2026", "evidence": [ "lean/docs/066.md:11: Uniform Cartier sections, bounded klt complements, and the paper's real-boundary and companion conclusions are outside this supporting statement.", "lean/docs/066.md:5: - [Uniform Cartier sections for Fano type contractions](../../preprints/Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026/Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026.pdf)", "corpus/openai-math/release.json:6807: This proves the finite-rational-coefficient form of Shokurov's bounded-klt-complement conjecture.", "lean/ComparatorChallenges/README.md:14: These setups verify supporting results, not the corresponding papers’ main", "lean/ComparatorChallenges/README.md:19: - `CartierChartCompactnessSupport.json`: compactness of normalized chart-field", "lean/ComparatorChallenges/README.md:20: retractions." ] }, { "clause": "For ϵ-lc Fano contractions over any algebraically closed characteristic-zero field, it gives klt complements near every base point, with index bounded only by dimension and positive rational ϵ.", "status": "declared", "coveredBy": [], "manuscript": "Bounded-klt-complements-for-Fano-contractions-September-25-2026", "evidence": [ "lean/docs/066.md:11: bounded klt complements, and the paper's real-boundary and companion conclusions are outside this supporting statement.", "corpus/openai-math/release.json:6807: admits, near each closed base point, a klt complement of index bounded only by d and ϵ." ] } ], "gap": "The whole headline. The challenge proves only that, on a closed étale chart (local Noetherian domain A, essentially of finite type and formally unramified over the localization R of k[σ] at the origin, formally étale over k[σ], k algebraically closed), the family of chart-field retraction valuations K → WithTop ℝ indexed by weights w ≥ 0 with Σ aᵢwᵢ = 1 (all aᵢ > 0) is compact in the pointwise topology.", "scopeNoteOnGap": "lean/docs/066.md:11: \"Uniform Cartier sections, bounded klt complements, and the paper's real-boundary and companion conclusions are outside this supporting statement.\"; README declares it a supporting-result comparison (README.md:19–20).", "note": "The family's only challenge, CartierChartCompactnessSupport, is on the README's list of supporting results (README.md:19-20), so SUPPORT-ONLY; the scope note also names bounded klt complements as outside (066.md:11), so both clauses are declared even though their manuscript (Bounded-klt-complements) is not linked. Flag: STATEMENT IMPORTS THE SOLUTION LIBRARY (CartierChartCompactnessSupport.lean:1-2 import OAI.AlgebraicGeometry.CartierSections.*); no definition hole.", "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "CartierChartCompactnessSupport", "flag": "IMPORTED DEFINITIONS: the challenge imports OAI.AlgebraicGeometry.CartierSections.FieldTopology and .FieldRetraction, so its statement closure is ~12 files of OpenAI's own library (FieldRetraction, ChartRetraction, ClosedCharts, ChartCompletion, InfinitesimalCharts, ExpansionValuation, FormalValuation, PolynomialValuation, Weights, ConeTopology, FractionValuation, FieldTopology) rather than the challenge file; I read the defining clauses of closedChartFieldRetraction, closedChartRetraction, closedChartExpansion, chartSeriesExpansion, expansionValuation, FormalMonomial.value/valuation, fractionValuation and normalizedWeights — genuine definitions, and grep finds no sorry/axiom/opaque/implemented_by in these files" }, { "challenge": "CartierChartCompactnessSupport", "flag": "VACUITY: none — the instance hypotheses are satisfiable (A = R itself)" } ], "flagReadings": null, "readings": [ "first/batch6.json", "second/second1.json" ] }, { "family": "071", "title": "Koebe’s circle-domain conjecture", "headline": "Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establishing this direction of the He–Schramm conjecture.", "challenges": [ "KoebeCircleDomains" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points.", "coveredBy": [ "KoebeCircleDomains" ], "note": "koebe_circle_domain: every open connected U in OnePoint ℂ admits a circle domain V and a conformal equivalence U → V." }, { "clause": "It also proves that circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establishing this direction of the He–Schramm conjecture.", "coveredBy": [ "KoebeCircleDomains" ], "note": "removability_implies_rigidity: circle domains U, V, frontier U conformally removable, f a conformal equivalence U → V ⇒ f agrees on U with a Möbius map." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "KoebeCircleDomains", "flag": "Cosmetic: the declarations sit in namespace OAI.Problem047 although the family is 071 (no effect on content)." } ], "flagReadings": null, "readings": [ "first/batch3.json" ] }, { "family": "072", "title": "Brennan's conjecture and the integral-means spectrum", "headline": "Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, $`|\\phi'|^s`$ is area-integrable for $`4/3\\lt s\\lt 4`$. The sharp universal integral-means identity is $`B_{\\mathcal S}(t)=|t|-1`$ for t ≤ −2. A strict bound $`B_b(-1)\\lt 1/4`$ for bounded univalent functions disproves Kraetzer's prediction at that parameter.", "challenges": [ "Brennan", "BrennanSharp", "StrictMeans" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, $`|\\phi'|^s`$ is area-integrable for $`4/3\\lt s\\lt 4`$.", "status": "stated", "coveredBy": [ "Brennan" ], "evidence": [ "lean/ComparatorChallenges/Brennan.lean:38: (∀ W : Set ℂ, IsOpen W → IsConnected W → SimplyConnectedSpace W →", "lean/ComparatorChallenges/Brennan.lean:40: DifferentiableOn ℂ φ W → Set.BijOn φ W disk →", "lean/ComparatorChallenges/Brennan.lean:41: ∀ s : ℝ, 4 / 3 < s → s < 4 →", "lean/ComparatorChallenges/Brennan.lean:42: IntegrableOn (fun z => ‖deriv φ z‖ ^ s) W volume) ∧" ], "note": "The extra hypothesis SphericalBoundaryNontrivial W is redundant: a holomorphic bijection onto the disk forces W ≠ ℂ (Liouville), and a simply connected W ≠ ℂ has more than one boundary point on the sphere. BrennanSharp adds endpoint sharpness via the Koebe map." }, { "clause": "The sharp universal integral-means identity is $`B_{\\mathcal S}(t)=|t|-1`$ for t ≤ −2.", "status": "undeclared", "coveredBy": [ "Brennan" ], "manuscript": "Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026", "evidence": [ "lean/ComparatorChallenges/Brennan.lean:36: integralMean f (-2) r ≤ C * (1 - r) ^ (-1 - ε)) ∧", "lean/ComparatorChallenges/Brennan.lean:37: spectrum (-2) = 1 ∧", "lean/docs/072.md:5: - [Brennan's conjecture and sharp inverse-square integral means](../../preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf)", "lean/docs/072.md:12: It also gives the uniform inverse-square radial-mean bound with exponent $-1-\\varepsilon$ for every $\\varepsilon>0$ and the spectrum value $B_{\\mathcal S}(-2)=1$.", "preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/build/sections/06-integrability.tex:47: \\begin{corollary}[The full negative linear tail]\\label{cor:negative-linear-tail}", "preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/build/sections/06-integrability.tex:56: B_{\\SSS}(t)=|t|-1\\qquad(t\\leq-2).", "preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/build/sections/06-integrability.tex:61: Put $a=|t|-2\\geq0$. The distortion lower bound" ], "note": "Only t = −2 is stated. The identity for t < −2 follows from the stated t = −2 bound only through the Koebe distortion theorem (|f'(z)| ≥ (1−|z|)/(1+|z|)³, not formalized here or in Mathlib) plus the Koebe-map lower bound — exactly the manuscript's corollary proof. The linked manuscript claims the full range; docs/072.md:12 describes the t = −2 part and never names the omission." }, { "clause": "A strict bound $`B_b(-1)\\lt 1/4`$ for bounded univalent functions disproves Kraetzer's prediction at that parameter.", "status": "stated", "coveredBy": [ "StrictMeans" ], "evidence": [ "lean/ComparatorChallenges/StrictMeans.lean:39: boundedSpectrum (-1) < (1 / 4 : EReal) ∧", "lean/ComparatorChallenges/StrictMeans.lean:40: (¬ ∀ p : ℝ, boundedSpectrum p = kraetzerPrediction p) := by", "lean/ComparatorChallenges/StrictMeans.lean:31: if |p| ≤ 2 then ((p ^ (2 : ℕ) / 4 : ℝ) : EReal)" ], "note": "kraetzerPrediction (−1) = 1/4, so boundedSpectrum (−1) < 1/4 is the failure at that parameter; B_b is taken over normalized bounded univalent maps (normalization does not change the growth exponent)." } ], "gap": "The headline claims B_S(t) = |t| − 1 for every t ≤ −2. Lean states it only at t = −2 (Brennan.lean:37). The manuscript states the full range as a corollary (sections/06-integrability.tex:56), so the headline matches the paper and the Lean is narrower by that corollary. Mathematically the gap is a short classical step (B_S(t)+t is nondecreasing by the distortion theorem, and the Koebe function gives the lower bound), but it is not stated.", "scopeNoteOnGap": "silent on t < −2. The nearest text describes only t = −2: lean/docs/072.md:12: \"It also gives the uniform inverse-square radial-mean bound with exponent $-1-\\varepsilon$ for every $\\varepsilon>0$ and the spectrum value $B_{\\mathcal S}(-2)=1$.\"", "note": "Brennan and the Kraetzer clause are stated; the negative linear tail B_S(t) = |t|−1 is stated only at t = −2. The full range is a corollary in a manuscript the note links, and the note describes only t = −2 without naming the omission. Agrees with the first reader.", "firstSuspicious": "The omitted range t < −2 is a short corollary in the manuscript; a second reader may judge it an honest scoping choice, but neither the scope note nor the challenge says it is left out, and the CONTENTS headline states the full range.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "Brennan", "flag": "VACUITY? checked, none: SphericalBoundaryNontrivial is automatically true for simply connected W ≠ ℂ, and W ≠ ℂ follows from a holomorphic bijection onto the disk (Liouville); the hypothesis is redundant, not restrictive." }, { "challenge": "Brennan", "flag": "JUNK? checked, none: for schlicht f and r < 1 the integrand |f'|^t is continuous and positive on the circle, so the interval integral and the log are genuine." }, { "challenge": "BrennanSharp", "flag": "JUNK? checked, none: ¬IntegrableOn could hold for a non-measurable function, but deriv of any function is measurable, so it means infinite integral." } ], "flagReadings": null, "readings": [ "first/batch5.json", "second/second2.json" ] }, { "family": "073", "title": "The Falconer distance conjecture", "headline": "Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set $`E\\subset\\mathbb R^d`$ with Hausdorff dimension greater than $`d/2`$ determines a set of Euclidean distances of positive Lebesgue measure.", "challenges": [ "PlanarFalconer", "FalconerAllDimensions" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set $`E\\subset\\mathbb R^d`$ with Hausdorff dimension greater than $`d/2`$ determines a set of Euclidean distances of positive Lebesgue measure.", "coveredBy": [ "FalconerAllDimensions", "PlanarFalconer" ], "note": "∀ d ≥ 2, compact E ⊆ ℝ^d (EuclideanSpace), d/2 < dimH E (in ℝ≥0∞) ⇒ Lebesgue measure of {dist x y : x,y ∈ E} > 0. PlanarFalconer is the d = 2 instance." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch3.json" ] }, { "family": "075", "title": "The $`L\\log L`$ Fourier-convergence conjecture", "headline": "Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in $`L\\log L(\\mathbb T)`$ converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the $`L\\log L`$ scale.", "challenges": [ "FourierLLogL" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in $`L\\log L(\\mathbb T)`$ converge almost everywhere along the full sequence", "coveredBy": [ "FourierLLogL" ], "note": "Sum over k = -N..N of fourierCoeff f k * fourier k x tends to f x outside a measurable null set, for every N in N (the full sequence)." }, { "clause": "This resolves the classical sufficiency conjecture at the $`L\\log L`$ scale", "coveredBy": [ "FourierLLogL" ], "note": "Same theorem." } ], "gap": null, "scopeNoteOnGap": "no gap", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "076", "title": "Real ultraflat Littlewood polynomials and unbounded binary merit factors", "headline": "Constructs polynomials with N consecutive coefficients in $`\\{-1,1\\}`$ whose modulus is $`(1+o(1))\\sqrt N`$ uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.", "challenges": [ "AsymptoticallyMinimalLittlewood", "LittlewoodFiniteFlatness" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Constructs polynomials with N consecutive coefficients in $`\\{-1,1\\}`$ whose modulus is $`(1+o(1))\\sqrt N`$ uniformly on the entire unit circle, for every sufficiently large integer length N.", "status": "unlinked", "coveredBy": [ "AsymptoticallyMinimalLittlewood" ], "manuscript": "Ultraflat-real-Littlewood-polynomials-October-5-2026", "evidence": [ "lean/ComparatorChallenges/AsymptoticallyMinimalLittlewood.lean:16: ∀ z : ℂ, ‖z‖ = 1 → ‖littlewoodValue ε z‖ ≤ (1 + η) * Real.sqrt N", "preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/build/sections/introduction.tex:20: (1-\\varepsilon)\\sqrt N", "preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/build/sections/introduction.tex:21: \\le\\left|\\sum_{k=0}^{N-1}\\varepsilon_k z^k\\right|", "preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026/build/introduction.tex:36: It gives no uniform lower bound for $|P(z)|$ and hence does", "preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026/build/introduction.tex:37: not assert two-sided uniform ultraflatness.", "lean/docs/076.md:5: - [Asymptotically minimal maxima of real Littlewood polynomials]" ] }, { "clause": "Thus real Littlewood polynomials are ultraflat, including at the real endpoints.", "status": "unlinked", "coveredBy": [], "manuscript": "Ultraflat-real-Littlewood-polynomials-October-5-2026", "evidence": [ "preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/build/sections/introduction.tex:13: can be ultraflat, through every sufficiently large integer length.", "preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/build/sections/introduction.tex:28: same polynomial on the entire circle, including the real points", "lean/docs/076.md:9: whose maximum modulus on the unit circle is at most $(1+\\eta)\\sqrt N$ for any fixed $\\eta>0$.", "lean/docs/076.md:1: # Real ultraflat Littlewood polynomials and unbounded binary merit factors" ] }, { "clause": "Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.", "status": "stated", "coveredBy": [ "AsymptoticallyMinimalLittlewood", "LittlewoodFiniteFlatness" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/AsymptoticallyMinimalLittlewood.lean:14: ∀ η : ℝ, 0 < η → ∃ N₀ : ℕ, 1 ≤ N₀ ∧ ∀ N : ℕ, N₀ ≤ N →", "preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026/build/merit-morse.tex:39: Parseval, and the elementary fourth-moment" ] } ], "gap": "The lower half of ultraflatness — min over |z| = 1 of |P_N(z)| ≥ (1−ε)√N for every large N, including z = ±1 — is not stated; the Lean gives the upper bound and finite-p L^p flatness only. The merit-factor clause is implied, not stated.", "scopeNoteOnGap": "silent — the note links only the September 23 paper and describes only the upper bound and L^p flatness: lean/docs/076.md:9: whose maximum modulus on the unit circle is at most $(1+\\eta)\\sqrt N$ for any fixed $\\eta>0$ / lean/docs/076.md:11: for every fixed finite $p>0$, the $L^p$ mean of", "note": "Merit factor appears in no Lean file, but the clause follows from AsymptoticallyMinimalLittlewood by an elementary, Mathlib-formalized step (named encoding/derivation): with F = N²/(‖P‖₄⁴ − N²) (Parseval for the aperiodic autocorrelations, orthonormality of the characters) and ‖P‖₄⁴ ≤ ‖P‖∞²‖P‖₂² ≤ (1+η)²N², F ≥ 1/((1+η)²−1) → ∞ through all lengths; LittlewoodFiniteFlatness at p = 4 gives it too. The two-sided (ultraflat) clauses come only from the October 5 'Ultraflat' manuscript, which the note does not link, and the one linked paper says in words that it does not assert them; so NARROWER — UNLINKED. I differ from the first reader only by the amendment-3 word (the first reader's gap is the same: the uniform lower bound).", "firstSuspicious": "Pattern seen across this batch: the CONTENTS.md headline was widened to manuscripts the scope note does not link, and the scope note never says those manuscripts are unformalized. Strictly that is silence, hence UNDECLARED; a second reader should weigh whether the scope note's paper list counts as a declaration. Here: the October 5 'Ultraflat' and 'Nearly-minimal maxima and positive minima' papers carry the headline's lower-bound clause.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json", "second/second3.json" ] }, { "family": "081", "title": "Riesz transforms and rectifiability in higher codimension", "headline": "Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and $`2\\le n\\le d-2`$, an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable whenever its n-dimensional Riesz transform is uniformly L2-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds.", "challenges": [ "RieszQuantitative" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable whenever its n-dimensional Riesz transform is uniformly L2-bounded over all positive hard truncations", "coveredBy": [ "RieszQuantitative" ], "note": "d ≥ 4, 2 ≤ n, n+2 ≤ d; regular (Radon) μ, AD-regular with constant C_AD at admissible radii, all hard truncations bounded by one C_R on L²(μ) ⇒ big pieces of Lipschitz images." }, { "clause": "The rectifiability bounds depend only on dimension, regularity and operator bounds.", "coveredBy": [ "RieszQuantitative" ], "note": "Quantifier order ∀ d n C_AD C_R ∃ θ M ∀ μ." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "082", "title": "Annular variation and dyadic absolute bounds for the triangular Hilbert transform", "headline": "Proves maximal and annular r-variation bounds, for every r > 2, from complex $`L^3(\\mathbb R^2)\\times L^3(\\mathbb R^2)`$ to $`L^{3/2}(\\mathbb R^2)`$. The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $`L^3\\times L^3\\times L^3`$ point.", "challenges": [ "TriangularHilbert" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Proves maximal and annular r-variation bounds, for every r > 2, from complex $`L^3(\\mathbb R^2)\\times L^3(\\mathbb R^2)`$ to $`L^{3/2}(\\mathbb R^2)`$.", "part": "the maximal bound", "status": "stated", "coveredBy": [ "TriangularHilbert" ], "evidence": [ "lean/ComparatorChallenges/TriangularHilbert.lean:45: maximalNorm F G ≤ (C : ℝ≥0∞) * eLpNorm F 3 volume * eLpNorm G 3 volume" ], "encoding": [ "L^{3/2} norm written as (∫⁻ maximal^(3/2))^(2/3)", "maximal set to 0 off GoodPoint, which is required to hold a.e." ] }, { "clause": "Proves maximal and annular r-variation bounds, for every r > 2, from complex $`L^3(\\mathbb R^2)\\times L^3(\\mathbb R^2)`$ to $`L^{3/2}(\\mathbb R^2)`$.", "part": "the annular r-variation bounds, every r > 2", "status": "unlinked", "coveredBy": [], "manuscript": "Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026", "evidence": [ "CONTENTS.md:2029: We prove the annular r-variation estimate for the triangular Hilbert transform from complex $`L^3\\times L^3`$ to L3/2 for every r > 2.", "lean/docs/082.md:5: - [The maximal triangular Hilbert transform at the symmetric point](../../preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/paper.pdf)" ] }, { "clause": "The maximal estimate controls both hard truncation endpoints", "status": "stated", "coveredBy": [ "TriangularHilbert" ], "evidence": [ "lean/ComparatorChallenges/TriangularHilbert.lean:35: ⨆ E : Endpoints, ENNReal.ofReal ‖truncation F G E z‖ else 0" ] }, { "clause": "and gives almost-everywhere and norm convergence", "status": "undeclared", "coveredBy": [], "manuscript": "The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026", "evidence": [ "CONTENTS.md:2047: The estimate yields joint almost-everywhere and L3/2 convergence as the lower endpoint tends to zero and the upper endpoint tends to infinity.", "preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/build/sections/consequences.tex:72: \\begin{theorem}[Joint bilinear principal values]\\label{thm:bilinear-pv}", "preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/build/sections/consequences.tex:78: exists almost everywhere and in $L^{3/2}(\\R^2)$, with", "lean/docs/082.md:9: It also establishes a common full-measure set on which the truncated integrals are defined and almost-everywhere measurability of the maximal output.", "lean/ComparatorChallenges/TriangularHilbert.lean:43: (∀ᵐ z ∂volume, GoodPoint F G z) ∧" ], "why": "MainEstimate states no limit of truncations. Convergence follows from main_estimate only with Banach's principle plus convergence on a dense class (compactly supported smooth inputs), and dominated convergence for the L^{3/2} limit, none of which is in the statement. The note describes the a.e. definedness and measurability it does cover and never says the principal value is left out." }, { "clause": "Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $`L^3\\times L^3\\times L^3`$ point.", "status": "stated", "coveredBy": [ "TriangularHilbert" ], "evidence": [ "preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/build/sections/consequences.tex:161: H\\\"older's inequality in $(x,y)$ bounds the absolute integral by", "preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/build/sections/consequences.tex:165: Theorem~\\ref{thm:main} gives its endpoint-uniform bound, and" ], "why": "The endpoint-uniform trilinear bound sup_{ε 2), the a.e. and norm convergence of the truncations, and the trilinear L^3 x L^3 x L^3 estimate. Also not stated: the dyadic triangular Hilbert form bound of the family's second manuscript, which the family title names but the headline text does not.", "scopeNoteOnGap": "silent. docs/082.md:9 describes only the maximal estimate (\"The formalized result proves the symmetric maximal estimate for the triangular Hilbert transform: ...\") and links only the maximal paper (line 5). It does not say that the variation, convergence or trilinear clauses are left out.", "note": "Only the maximal bound (both endpoints) is stated. The annular r-variation bound comes from the unlinked October 5 paper (UNLINKED), but the a.e. and L^{3/2} convergence is the linked paper's own Theorem (bilinear-pv), claimed in its abstract and the headline, absent from the statement, and not named by the note, so the family is NARROWER — UNDECLARED. Agree with the first reader on the word; I read the pairing clause as stated (Hölder + Fubini from the maximal bound, as in the paper), where the first reader did not, which does not change the word.", "firstSuspicious": "UNDECLARED rests on silence. The scope note describes the formalized maximal estimate precisely and names only the maximal paper, but no sentence says that the variation, convergence and trilinear results are outside.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "TriangularHilbert", "flag": "NARROWER — UNDECLARED: only the maximal estimate is stated. Lean side: lines 35, 40-45. Headline side: the variation, convergence and trilinear clauses. Scope note side: docs/082.md:9 is silent on what is left out." } ], "flagReadings": null, "readings": [ "first/batch7.json", "second/second4.json" ] }, { "family": "083", "title": "Hilbert transforms along Lipschitz directions", "headline": "Proves a uniform strong L2 bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant. The estimate is uniform over inner truncations and yields an L2-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale.", "challenges": [ "LipschitzHilbert" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves a uniform strong L2 bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant.", "coveredBy": [ "LipschitzHilbert" ], "note": "uniform_truncation: ∃ C ∀ K > 0 ∀ 0 < ε ≤ R ≤ 1/(10⁶K) ∀ K-Lipschitz unit v ∀ Schwartz f, ‖H_{ε,R} f‖₂ ≤ C‖f‖₂." }, { "clause": "The estimate is uniform over inner truncations", "coveredBy": [ "LipschitzHilbert" ], "note": "C independent of ε in both theorems." }, { "clause": "yields an L2-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale", "coveredBy": [ "LipschitzHilbert" ], "note": "main: for 1-Lipschitz unit fields at one fixed scale a ∈ (0, 1/2): principal-value L² bound, weak (2,2) bound and bounded L² extensions; the K-Lipschitz version follows by dilation (not stated) — the scope note names the 1-Lipschitz normalisation." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "084", "title": "The geometric case of the Erdős similarity conjecture", "headline": "For every fixed $`q\\in(0,1)`$, constructs compact subsets of $`[0,1]`$ with measure arbitrarily close to one containing no translated and nontrivially dilated copy of $`\\{q^n:n\\ge1\\}`$, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.", "challenges": [ "DyadicAvoidance" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "For every fixed $`q\\in(0,1)`$, constructs compact subsets of $`[0,1]`$ with measure arbitrarily close to one containing no translated and nontrivially dilated copy of $`\\{q^n:n\\ge1\\}`$, with dilations of either sign.", "part": "q = 1/2", "status": "stated", "coveredBy": [ "DyadicAvoidance" ], "reading": "∀ η ∈ (0,1) ∃ compact E ⊆ [0,1], vol E > 1 − η, ∀ x ∀ s ≠ 0 ∃ n ≥ 1, x + s·2⁻ⁿ ∉ E (both signs of s).", "evidence": [ "lean/ComparatorChallenges/DyadicAvoidance.lean:10: (2 : ℝ)⁻¹ ^ n", "lean/ComparatorChallenges/DyadicAvoidance.lean:15: MeasureTheory.volume E > ENNReal.ofReal (1 - η) ∧", "lean/ComparatorChallenges/DyadicAvoidance.lean:16: ∀ x s : ℝ, s ≠ 0 → ∃ n : ℕ, 1 ≤ n ∧" ] }, { "clause": "For every fixed $`q\\in(0,1)`$, constructs compact subsets of $`[0,1]`$ with measure arbitrarily close to one containing no translated and nontrivially dilated copy of $`\\{q^n:n\\ge1\\}`$, with dilations of either sign.", "part": "every q ∈ (0,1) other than 1/2", "status": "unlinked", "coveredBy": [], "manuscript": "The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026", "evidence": [ "lean/ComparatorChallenges/DyadicAvoidance.lean:9: def dyadicPoint (n : ℕ) : ℝ :=", "lean/ComparatorChallenges/DyadicAvoidance.lean:10: (2 : ℝ)⁻¹ ^ n", "CONTENTS.md:2079: For every fixed $`q\\in(0,1)`$ and every $`\\eta\\in(0,1)`$, we construct a compact set $`E_{q,\\eta}\\subseteq[0,1]`$ of measure greater than $`1-\\eta`$ containing no nontrivial affine copy of $`\\{q^n:n\\ge1\\}`$", "lean/docs/084.md:5: - [The dyadic case of the Erdős similarity conjecture](../../preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026/paper.pdf)", "lean/docs/084.md:9: The formalization proves the dyadic case of the Erdős similarity conjecture." ] }, { "clause": "This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.", "status": "unlinked", "coveredBy": [], "manuscript": "The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026", "evidence": [ "CONTENTS.md:2079: We prove the geometric-progression case of the Erdős similarity conjecture.", "lean/docs/084.md:5: - [The dyadic case of the Erdős similarity conjecture](../../preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026/paper.pdf)", "lean/docs/084.md:9: The formalization proves the dyadic case of the Erdős similarity conjecture." ] } ], "gap": "The headline claims every ratio q ∈ (0,1); the Lean states only q = 1/2. The general-ratio theorem is the October 5 manuscript ('The geometric case…'), which the scope note does not link.", "scopeNoteOnGap": "silent on the omission. The note says lean/docs/084.md:9: \"The formalization proves the dyadic case of the Erdős similarity conjecture.\" and links only the September 25 dyadic manuscript (084.md:5); no sentence says the general-ratio case of the headline is left out. Declared, at most, by restriction.", "note": "Only q = 1/2 is stated (exactly, both signs of the dilation). Every other ratio comes from the October-5 'geometric case' manuscript, which docs/084.md does not link (it links only the September-25 dyadic paper); the note's 'dyadic case' sentence describes the formalized part and does not name the omission, and its title alone declares nothing. First reader: UNDECLARED, recorded before UNLINKED existed; same facts.", "firstSuspicious": "DECLARED vs UNDECLARED hinges on whether naming the narrower case ('the dyadic case') counts as saying what is left out; I applied the literal rule (the note must say what is left out). Same pattern as 186, 197, 247: the headline was rewritten around an October 4–5 manuscript that the scope note does not link, while the Lean and the note still describe the September result.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch6.json", "second/second5.json" ] }, { "family": "085", "title": "Endpoint Sobolev regularity of centered disk averages", "headline": "Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real $`f\\in W^{1,1}(\\mathbb R^2)`$, the centered disk maximal function satisfies $`\\|\\nabla Mf\\|_1\\le C\\|\\nabla f\\|_1`$ with an absolute constant. It belongs locally to $`W^{1,1}`$ and has a globally integrable weak gradient.", "challenges": [ "SignedFiniteBand", "DiskMaximal" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem.", "coveredBy": [ "DiskMaximal" ], "note": "Via the endpoint statement below." }, { "clause": "For every real $`f\\in W^{1,1}(\\mathbb R^2)`$, the centered disk maximal function satisfies $`\\|\\nabla Mf\\|_1\\le C\\|\\nabla f\\|_1`$ with an absolute constant.", "coveredBy": [ "DiskMaximal" ], "note": "∃ C ≥ 0 before ∀ f, g: absolute constant; f ∈ W^{1,1} encoded as Integrable f, Integrable g, g a weak gradient of f." }, { "clause": "It belongs locally to $`W^{1,1}`$ and has a globally integrable weak gradient.", "coveredBy": [ "DiskMaximal" ], "note": "InW11Loc (maximalReal f) and ∃ G integrable weak gradient; also a.e. finiteness of Mf." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "DiskMaximal", "flag": "JUNK checked, none: the maximal function is an ℝ≥0∞ supremum; the Bochner integrals in HasWeakGradient are of integrable functions (f, g by hypothesis; maximalReal f locally integrable by the conclusion, times a bounded compactly supported test derivative), so none is a junk zero" } ], "flagReadings": null, "readings": [ "first/batch6.json" ] }, { "family": "087", "title": "The Mahler conjectures, functional inequalities and polar-product symplectic width", "headline": "Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product $`K\\times K^\\circ`$ in dimension $`2n`$ has Gromov width 4.", "challenges": [ "MahlerConjecture", "SymmetricMahlerEquality", "GeneralMahler", "SymmetricPolar" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified.", "coveredBy": [ "MahlerConjecture", "SymmetricMahlerEquality", "GeneralMahler" ], "note": "symmetric inequality |K||K°| >= 4^n/n! for n>=1; equality iff invertible linear image of a Hanner body; general: inf over interior centers z of |K||(K-z)°| >= (n+1)^(n+1)/(n!)^2 with equality iff K is a simplex" }, { "clause": "The corresponding sharp functional Mahler inequalities also hold.", "coveredBy": [], "note": "not stated in any challenge; declared out by the scope note" }, { "clause": "For n ≥ 2, every symmetric polar product $`K\\times K^\\circ`$ in dimension $`2n`$ has Gromov width 4.", "coveredBy": [ "SymmetricPolar" ], "note": "Gromov width of int K × int K° equals 4 and every ball of capacity c<4 embeds, capacity normalized as π r^2" } ], "gap": "the sharp functional Mahler inequalities", "scopeNoteOnGap": "The nonsymmetric Mahler conjecture and the functional inequalities are not included. (docs/087.md:13) and The paper's functional inequality is outside this selected statement. (docs/087.md:17). Note: line 13's 'nonsymmetric Mahler conjecture ... not included' is contradicted by lines 15-17 and by GeneralMahler.lean, which does state it — a stale sentence in the scope note, not a gap in the Lean.", "note": null, "firstSuspicious": "docs/087.md:13 says the nonsymmetric Mahler conjecture is not included while lines 15-17 and GeneralMahler.lean include it — internal inconsistency in OpenAI's scope note (no effect on the word).", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "GeneralMahler", "flag": "JUNK? cleared: volumeProduct is an sInf of reals over a nonempty set bounded below by 0, and the polar at an interior point is bounded so its volume is finite" } ], "flagReadings": [ { "challenge": "GeneralMahler", "flag": "JUNK? cleared: volumeProduct is an sInf of reals over a nonempty set bounded below by 0, and the polar at an interior point is bounded so its volume is finite", "decision": "cleared", "why": "volumeProduct is Real.sInf over the image of interior K, which is nonempty by hinterior, of values ≥ 0. It is therefore a genuine infimum. Both toReal's are faithful. K is compact, so vol K < ∞. For z ∈ interior K there is r > 0 with closedBall z r ⊆ K, and ⟪y, x − z⟫ ≤ 1 for all x ∈ K forces ‖y‖ ≤ 1/r. So polarAt K z ⊆ closedBall 0 (1/r), which has finite volume. A junk 0 would make the lower bound false rather than easier. The minimisation over interior centres is the standard nonsymmetric volume product (the Santaló point).", "evidence": [ "lean/ComparatorChallenges/GeneralMahler.lean:16: {y | ∀ x ∈ K, ⟪y, x - z⟫ ≤ 1}", "lean/ComparatorChallenges/GeneralMahler.lean:20: sInf ((fun z => (volume K).toReal * (volume (polarAt K z)).toReal) ''", "lean/ComparatorChallenges/GeneralMahler.lean:21: interior K)", "lean/ComparatorChallenges/GeneralMahler.lean:26: (hcompact : IsCompact K) (hconvex : Convex ℝ K)", "lean/ComparatorChallenges/GeneralMahler.lean:27: (hinterior : (interior K).Nonempty) :", "lean/ComparatorChallenges/GeneralMahler.lean:28: ((n : ℝ) + 1) ^ (n + 1) / (Nat.factorial n : ℝ) ^ 2 ≤ volumeProduct K ∧" ] } ], "readings": [ "first/batch8.json", "flags/flags2.json" ] }, { "family": "088", "title": "Sharp projection-body inequalities and a counterexample to simplex maximization", "headline": "Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension.", "challenges": [ "PettyProjectionVolume", "ProjectionCounterexample", "ProjectionVolume" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume.", "coveredBy": [ "PettyProjectionVolume" ], "note": "For every n ≥ 4 and convex body K: κ_{n−1}^n κ_n^{2−n} ≤ |ΠK|/|K|^{n−1}, with equality iff K is an ellipsoid." }, { "clause": "Also establishes the full Lutwak–Petty projection inequalities.", "coveredBy": [], "note": "Not stated." }, { "clause": "In contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension.", "coveredBy": [ "ProjectionCounterexample", "ProjectionVolume" ], "note": "Covered only in dimension 20, by the factor 22355476/22020096 ≈ 1.015; 'exponential factor in every sufficiently large dimension' is not stated." } ], "gap": "The Lutwak–Petty projection inequalities, and the exponential-factor counterexample in every sufficiently large dimension (only n = 20 is stated).", "scopeNoteOnGap": "lean/docs/088.md:16: \"Other inequalities in the projection-body family are not included.\"; lean/docs/088.md:22: \"The paper's exponential-factor result for every sufficiently large dimension is not included.\"", "note": null, "firstSuspicious": "'Lutwak–Petty projection inequalities' are declared only through the generic 'Other inequalities in the projection-body family are not included'; I counted that as declared.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "ProjectionCounterexample", "flag": "JUNK checked: a degenerate shadow measure (0 or ∞) would make every ratio 0 and the theorem false, not trivial; the simplex ratio is a genuine positive real" } ], "flagReadings": null, "readings": [ "first/batch6.json" ] }, { "family": "089", "title": "Bounded-distortion L1 embeddings of planar and bounded-treewidth graphs", "headline": "Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture. Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real L1 with universal distortion for planar graphs, and distortion depending only on treewidth for bounded-treewidth graphs. The corresponding multicommodity flow–cut gaps are uniformly bounded.", "challenges": [ "PlanarL1", "BoundedTreewidthL1" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture.", "status": "stated", "coveredBy": [ "PlanarL1", "BoundedTreewidthL1" ], "evidence": [ "lean/ComparatorChallenges/PlanarL1.lean:34-44: def MainStatement : Prop := ⏎ ∃ C : ℝ, 1 ≤ C ∧ ⏎ ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (length : Fin n → Fin n → ℝ), ⏎ G.Connected → IsPlanar G → ⏎ (∀ u v, length u v = length v u) → ⏎ (∀ u v, G.Adj u v → 0 < length u v) → ⏎ ∃ (Ω : Type) (measurable : MeasurableSpace Ω) ⏎ (μ : @MeasureTheory.Measure Ω measurable) ⏎ (f : Fin n → @MeasureTheory.Lp Ω ℝ measurable _ 1 μ), ⏎ ∀ x y, graphDistance G length x y ≤ ‖f x - f y‖ ∧ ⏎ ‖f x - f y‖ ≤ C * graphDistance G length x y", "lean/ComparatorChallenges/BoundedTreewidthL1.lean:36-41: def MainStatement : Prop := ⏎ ∀ k : ℕ, 2 ≤ k → ∃ C : ℝ, 1 ≤ C ∧ ⏎ ∀ (V : Type) [Fintype V] [DecidableEq V] [Nonempty V] ⏎ (G : SimpleGraph V), ⏎ G.Connected → HasTreeDecomposition G k → ⏎ ∀ ℓ : G.edgeSet → ℝ, (∀ e, 0 < ℓ e) → HasL1Embedding G ℓ C" ] }, { "clause": "Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real L1 with universal distortion for planar graphs, and distortion depending only on treewidth for bounded-treewidth graphs.", "status": "stated", "coveredBy": [ "PlanarL1", "BoundedTreewidthL1" ], "encoding": "planarity as a topological drawing by injective arcs with disjoint interiors; treewidth as bag size ≤ k (k ≥ 2, i.e. treewidth ≤ k−1); planar target is L¹ of some measure space, treewidth target is finite-dimensional ℓ₁; distance = sInf of walk/path lengths (nonempty by connectivity, nonnegative)", "evidence": [ "lean/ComparatorChallenges/PlanarL1.lean:35-39: ∃ C : ℝ, 1 ≤ C ∧ ⏎ ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (length : Fin n → Fin n → ℝ), ⏎ G.Connected → IsPlanar G → ⏎ (∀ u v, length u v = length v u) → ⏎ (∀ u v, G.Adj u v → 0 < length u v) →", "lean/ComparatorChallenges/BoundedTreewidthL1.lean:37-41: ∀ k : ℕ, 2 ≤ k → ∃ C : ℝ, 1 ≤ C ∧ ⏎ ∀ (V : Type) [Fintype V] [DecidableEq V] [Nonempty V] ⏎ (G : SimpleGraph V), ⏎ G.Connected → HasTreeDecomposition G k → ⏎ ∀ ℓ : G.edgeSet → ℝ, (∀ e, 0 < ℓ e) → HasL1Embedding G ℓ C" ] }, { "clause": "The corresponding multicommodity flow–cut gaps are uniformly bounded.", "status": "undeclared", "coveredBy": [], "manuscript": "Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026 ; L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026", "classicalBridge": "follows from the stated embeddings only through the Linial–London–Rabinovich (1995) / Aumann–Rabani (1998) theorem that the flow–cut gap is at most the L1 distortion (the manuscripts cite Gupta–Newman–Rabinovich–Sinclair 2004, §3 Thm 3.2); not formalized, so not stated", "evidence": [ "preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/introduction.tex:60-62: graph~\\cite[Section~3, Theorem~3.2]{GNRS04}. Hence ⏎ Theorem~\\ref{thm:main} also gives a universal bound for the undirected ⏎ fractional flow--cut gap on planar graphs, with arbitrary demands.", "preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/build/01-introduction.tex:79-82: For graphs with bags of size at most $k$, Theorem~\\ref{thm:main} ⏎ therefore gives ⏎ \\[ ⏎ \\lambda_*\\leq\\phi_*\\leq C_k\\lambda_*.", "lean/docs/089.md:10: The planar case of the Gupta–Newman–Rabinovich–Sinclair embedding problem asks for a universal distortion bound in $L_1$. The formalized result gives one constant $C$ for every finite connected planar graph with arbitrary positive real edge lengths: its weighted shortest-path metric embeds into real $L_1$ with distortion at most $C$. The constant is independent of the number of vertices and the ratios between edge lengths; singleton graphs are included.", "lean/docs/089.md:12: The formalized result gives a uniform $L_1$ embedding bound for each bounded-treewidth class. For every bag-size bound $k\\ge2$, there is a constant $C(k)$ such that the weighted shortest-path metric of every nonempty finite connected graph with a tree decomposition of bag size at most $k$ embeds into finite-dimensional real $L_1$ with distortion at most $C(k)$. Edge lengths may be arbitrary positive real numbers, and the constant is independent of the graph and decomposition size.", "lean/ComparatorChallenges/PlanarL1.lean:43-44: ∀ x y, graphDistance G length x y ≤ ‖f x - f y‖ ∧ ⏎ ‖f x - f y‖ ≤ C * graphDistance G length x y", "lean/ComparatorChallenges/BoundedTreewidthL1.lean:30-34: def HasL1Embedding (ℓ : G.edgeSet → ℝ) (C : ℝ) : Prop := ⏎ ∃ m : ℕ, ∃ F : V → PiLp (1 : ENNReal) (fun _ : Fin m => ℝ), ⏎ ∀ u v : V, ⏎ shortestPathDistance G ℓ u v ≤ ‖F u - F v‖ ∧ ⏎ ‖F u - F v‖ ≤ C * shortestPathDistance G ℓ u v" ] } ], "gap": "Only the flow–cut-gap sentence, a classical corollary of the stated embeddings (LLR 1995 / Aumann–Rabani 1998); no flow or cut object appears in either challenge.", "scopeNoteOnGap": "silent (089.md:10 and 089.md:12 describe only the two L1-embedding statements; neither abstract mentions flow–cut gaps)", "note": "Both embedding theorems are stated as claimed. The flow–cut sentence is claimed in both linked papers (planar introduction.tex:60-62; treewidth 01-introduction.tex:79-82) but no challenge defines a flow or a cut and the note never mentions it; it is one classical theorem away (LLR95/AR98), which under the rule still makes it not stated. Agree with the first reader; the gap is small.", "firstSuspicious": "The word depends on whether a headline sentence that follows from the stated theorem by a single named classical theorem counts as stated. Under the literal rule it does not (NARROWER — UNDECLARED, gap = classical corollary only); under a 'classical bridge is an encoding step' rule the family is MATCHES. Same question for 326. Second reader should decide both together.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch6.json", "second/second6.json" ] }, { "family": "090", "title": "Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy", "headline": "Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk density one. It also minimizes unit-background renormalized Riesz energies for $`0\\lt s\\lt 2`$ and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy.", "challenges": [ "AtomicGaussian", "TriangularEnergy", "TriangularGaussian", "PlanarPacking" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk density one.", "status": "stated", "coveredBy": [ "TriangularEnergy" ], "evidence": [ "lean/ComparatorChallenges/TriangularEnergy.lean:66: theorem universal_energy_minimum (g : ℝ → ℝ) (C : Set Plane)", "lean/ComparatorChallenges/TriangularEnergy.lean:67: (hg : AdmissiblePotential g) (hC : LocallyFinite C) (hd : DensityOne C) :", "lean/ComparatorChallenges/TriangularEnergy.lean:68: latticeEnergy g ≤ energy g C ∧ latticeEnergy g = energy g A := by" ], "note": "Encoding: energies in ℝ≥0∞ (infinite energies included); lower energy per particle = liminf over radii of ordered-pair averages on closed disks at 0; complete monotonicity via (-1)^r g^(r) ≥ 0 on (0,∞) for a C^∞ g; A = covolume-one triangular lattice. AtomicGaussian and TriangularGaussian state the sharp Gaussian minorants the proof route uses; PlanarPacking (Cohn–Elkies sharpness) is outside the headline." }, { "clause": "It also minimizes unit-background renormalized Riesz energies for $`0\\lt s\\lt 2`$", "status": "unlinked", "coveredBy": [], "manuscript": "Universal-optimality-of-the-triangular-lattice-September-23-2026", "evidence": [ "CONTENTS.md:2227: [Universal optimality of the triangular lattice](preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/paper.pdf)", "CONTENTS.md:2229: We also prove triangular minimality for planar logarithmic and Riesz renormalized energies, with $`0\\lt s\\lt 2`$ in the Riesz case, and the corresponding jellium minima.", "lean/docs/090.md:5: - [An atomic certificate for triangular-lattice universal optimality](../../preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/paper.pdf)", "lean/docs/090.md:6: - [A sharp Fourier certificate for planar circle packing](../../preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf)" ], "note": "No challenge mentions a renormalized, Riesz, jellium or Coulomb energy." }, { "clause": "and Coulomb energy", "status": "unlinked", "coveredBy": [], "manuscript": "Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026", "evidence": [ "CONTENTS.md:2245: [Triangular minimality for planar Coulomb renormalized energy](preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/paper.pdf)", "CONTENTS.md:2247: We prove the Sandier–Serfaty conjecture: the triangular lattice of covolume one minimizes planar Coulomb renormalized energy over all admissible curl-free fields with a unit uniform background.", "lean/docs/090.md:5: - [An atomic certificate for triangular-lattice universal optimality](../../preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/paper.pdf)", "lean/docs/090.md:6: - [A sharp Fourier certificate for planar circle packing](../../preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf)" ], "note": "Also the logarithmic case of the (equally unlinked) universal-optimality manuscript." }, { "clause": "resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy.", "status": "unlinked", "coveredBy": [], "manuscript": "Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026", "evidence": [ "CONTENTS.md:2247: Combined with Bétermin and Sandier's asymptotic formula, this also proves the Brauchart–Hardin–Saff conjecture for the linear term of optimal ordered-pair logarithmic energy on the unit two-sphere.", "lean/docs/090.md:5: - [An atomic certificate for triangular-lattice universal optimality](../../preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/paper.pdf)", "lean/docs/090.md:6: - [A sharp Fourier certificate for planar circle packing](../../preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf)" ] } ], "gap": "Triangular minimality for unit-background renormalized Riesz energies (0Lp Brunn–Minkowski inequalities and the B-conjecture" ] } ], "gap": "The B-conjecture for every even log-concave Radon measure (the abstract derives it via Saroglou's transfer theorem and a support-subspace reduction) and the L_p Brunn–Minkowski inequality for 0 < p < 1 (derivable from log-BM by a classical argument) are not stated.", "scopeNoteOnGap": "silent (091.md:9 describes only the logarithmic inequality for Lebesgue volume; the B-conjecture and L_p are not mentioned)", "note": "Only the log-BM inequality for Lebesgue volume is stated. The B-conjecture for even log-concave measures needs Saroglou's transfer theorem and a support-subspace reduction, and the additive Lp-BM for 0 0, ∃ n₀, ∀ n ≥ n₀, each of translativeSup, symmetricTranslativeSup, latticeSup, symmetricLatticeSup lies in [c n log n, C n log n] (as ℝ≥0∞, so finiteness is part of the claim)." }, { "clause": "Every convex body in ℝn, n ≥ 2, admits a lattice covering of density at most $`Cn\\log n`$", "coveredBy": [ "SingleLatticeCovering" ], "note": "∃ C > 0 ∀ n ≥ 2 ∀ convex body K ⊆ ℝⁿ ∃ discrete full-rank ℤ-lattice L with K + L = ℝⁿ and vol(K)/covol(L) ≤ C n log n." }, { "clause": "centrally symmetric examples in every sufficiently large dimension require at least $`cn\\log n`$ even without the lattice restriction, for absolute $`c,C\\gt 0`$.", "coveredBy": [ "CoveringDensity" ], "note": "Lower bound c n log n ≤ symmetricTranslativeSup n for n ≥ n₀. A lower bound on a supremum yields, for any c' < c, an individual symmetric body with θ_T(K) > c' n log n, so the per-example form follows with a smaller absolute constant." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "CoveringDensity", "flag": "ENCODING (not a gap, mild second-reader item): θ_T uses the upper (limsup) centre intensity. A lower bound on an infimum of upper densities is a priori weaker than one on lower densities, but the two infima coincide by the standard periodisation of near-optimal coverings, so the stated Θ(n log n) is the usual one." } ], "flagReadings": null, "readings": [ "first/batch3.json" ] }, { "family": "094", "title": "Subpolynomial dimension reduction in Lp", "headline": "For every fixed $`1\\lt p\\lt \\infty`$ and distortion D > 1, every n-point subset of real Lp embeds into $`\\ell_p^d`$ with distortion at most D and dimension $`d=n^{o(1)}`$, answering Naor's sublinear-dimension question for p ≠ 2. In contrast, exact embeddings require worst-case dimension $`\\Theta(n^2)`$ when p ≠ 2.", "challenges": [ "SubpolynomialLp" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "For every fixed $`1\\lt p\\lt \\infty`$ and distortion D > 1, every n-point subset of real Lp embeds into $`\\ell_p^d`$ with distortion at most D and dimension $`d=n^{o(1)}`$", "coveredBy": [ "SubpolynomialLp" ], "note": "Covered for every p ∈ (1,∞) EXCEPT p = 2 (hypothesis hp2 : p ≠ 2). For p ≠ 2 and D > 1: GoodDimension holds at the least dimension, the dimension is ≤ exp(C (log n)^γ(p)) with γ(p) < 1, and log(dimension)/log n → 0. The p = 2 case (Johnson–Lindenstrauss, classical) is not stated." }, { "clause": "answering Naor's sublinear-dimension question for p ≠ 2", "coveredBy": [ "SubpolynomialLp" ], "note": "Follows from the first and third conjuncts for p ≠ 2." }, { "clause": "In contrast, exact embeddings require worst-case dimension $`\\Theta(n^2)`$ when p ≠ 2", "coveredBy": [ "SubpolynomialLp" ], "note": "Second conjunct: for n ≥ 9, ⌊(n−1)/4⌋² ≤ dimension(p,n,1) ≤ C(n,2), and log(dimension)/log n → 2 at D = 1." } ], "gap": "The headline's first sentence quantifies over every 1 < p < ∞; the Lean theorem assumes p ≠ 2, so the (classical, Johnson–Lindenstrauss) p = 2 case of the subpolynomial-dimension clause is not stated.", "scopeNoteOnGap": "docs/094.md:9: For $p>1$, $p\\ne2$, the formalization bounds the least dimension needed to embed every $n$-point subset of real $L_p$ with distortion $D$.…", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "095", "title": "Hyperbolicity cones without semidefinite lifts", "headline": "Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral.", "challenges": [ "HyperbolicCones" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows.", "status": "unlinked", "coveredBy": [], "manuscript": "Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026", "evidence": [ "lean/docs/095.md:5: - [A nonspectrahedral hyperbolicity cone](../../preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf)", "lean/ComparatorChallenges/HyperbolicCones.lean:73: ∀ (N : ℕ) (_hN : 0 < N) (L : Ambient →ₗ[ℝ] Sym N),", "lean/ComparatorChallenges/HyperbolicCones.lean:74: cone ≠ {x | (L x : Mat N ℝ).PosSemidef} := by", "CONTENTS.md:2336: We prove that not every hyperbolicity cone is a spectrahedral shadow: some closed hyperbolicity cones admit no finite affine semidefinite lift", "CONTENTS.md:2354: Thus, although this cone has no semidefinite representation in its original coordinates, it admits one when auxiliary variables are allowed." ], "note": "The challenge excludes only pencils in the original 23 coordinates; no lift or projection appears. The family's other October-5 manuscript says this very cone HAS an exact semidefinite lift, so the formalized example cannot witness this clause at all. The note's title (docs/095.md:1) names the October-5 result, but a title does not declare an omission and the note does not link that paper." }, { "clause": "The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used.", "status": "unlinked", "coveredBy": [], "manuscript": "Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026", "evidence": [ "lean/docs/095.md:5: - [A nonspectrahedral hyperbolicity cone]", "CONTENTS.md:2336: even with arbitrary real coefficients and any finite number of auxiliary variables." ], "note": "Same manuscript and same gap as the previous clause." }, { "clause": "This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral.", "status": "stated", "coveredBy": [ "HyperbolicCones" ], "evidence": [ "lean/ComparatorChallenges/HyperbolicCones.lean:71: MvPolynomial.eval (coordinates basePoint) polynomial = 1 ∧", "lean/ComparatorChallenges/HyperbolicCones.lean:72: (∀ (x : Ambient) (z : ℂ), (linePolynomial x).aeval z = 0 → z.im = 0) ∧", "lean/ComparatorChallenges/HyperbolicCones.lean:74: cone ≠ {x | (L x : Mat N ℝ).PosSemidef} := by" ], "note": "An explicit degree-20 form in 23 variables, hyperbolic in direction e with p(e) = 1, whose closed hyperbolicity cone is not {x | L x ⪰ 0} for any linear L into N×N symmetric matrices, N > 0. For a cone a linear pencil loses nothing. Degree 20 against the September abstract's 16 is an honest choice of representative: the manuscript's reduced polynomial has the same cone (build/sections/reduced-polynomial.tex). linePolynomial is never the zero polynomial (leading coefficient p(e) = 1), so the cone is not junk." } ], "gap": "the headline's lead claim (Projected Lax conjecture false; some hyperbolicity cones are not spectrahedral shadows / admit no lift with auxiliary variables) is not stated; only non-spectrahedrality in the original coordinates is", "scopeNoteOnGap": "silent — docs/095.md links only the September-24 nonspectrahedral paper and describes only: \"its closed hyperbolicity cone cannot be represented as the positive-semidefinite region of any finite real symmetric linear matrix pencil, in any positive matrix size\" (docs/095.md:9); it never mentions spectrahedral shadows, lifts, auxiliary variables or the Projected Lax conjecture", "note": "The generalized Lax clause is stated. The lead claim (the Projected Lax conjecture is false; no lift with auxiliary variables) comes from the October-5 manuscript, which the note does not link, and the formalized cone has a lift by the family's third manuscript. The first reader found the same gap and called it UNDECLARED before amendment 3.", "firstSuspicious": "The headline's lead clause is the October-5 'Hyperbolicity Cones Without Semidefinite Lifts' result, whose examples are different cones from the formalized one (the formalized cone has a lift per the other October-5 abstract). Re-read independently before stating on the page.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "HyperbolicCones", "flag": "VACUITY? cleared: N=0 excluded, and the hyperbolicity conjunct is stated, so the cone is a genuine closed hyperbolicity cone" } ], "flagReadings": null, "readings": [ "first/batch8.json", "second/second2.json" ] }, { "family": "096", "title": "The Gaussian propeller conjecture in every dimension", "headline": "Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most $`9/(8\\pi)`$. In dimension at least two, three planar sectors of angle $`2\\pi/3`$, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor $`(8\\pi/9)(1-1/k)`$ for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.", "challenges": [ "GaussianPropeller" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most $`9/(8\\pi)`$.", "coveredBy": [ "GaussianPropeller" ], "note": "every d>0, k>0, every k-tuple of measurable sets that partitions ℝ^d up to a γ-null set" }, { "clause": "In dimension at least two, three planar sectors of angle $`2\\pi/3`$, extended orthogonally, attain the bound.", "coveredBy": [ "GaussianPropeller" ], "note": "explicit 120° sectors in coordinates 0,1 (extra cells empty), for d>=2, k>=3" }, { "clause": "Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor $`(8\\pi/9)(1-1/k)`$ for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.", "coveredBy": [], "note": "not stated; declared out" } ], "gap": "the kernel-clustering NP-hardness consequence", "scopeNoteOnGap": "The separate Gaussian-maxima and kernel-clustering consequences are not included. (docs/096.md:9)", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "GaussianPropeller", "flag": "JUNK? cleared: x ↦ x is γ-integrable, so the Bochner centroids are genuine" } ], "flagReadings": [ { "challenge": "GaussianPropeller", "flag": "JUNK? cleared: x ↦ x is γ-integrable, so the Bochner centroids are genuine", "decision": "cleared", "why": "The only junk route is the Bochner integral in `centroid` returning 0 for a non-integrable integrand, which would LOWER `value` and so make the ≤ 9/(8π) bound of conjunct 1 easier. It does not arise: the identity map has finite first moment under the standard Gaussian on the finite-dimensional EuclideanSpace (all moments are finite), so it is integrable, hence integrable on every cell, and IsPartition requires every cell to be measurable, so `∫ x in A i` is a genuine restricted integral. In conjunct 2 a junk value would make the equality false, not true. Sanity check of the constant: each 120° sector has centroid norm √3·√(π/2)/(2π), squared 3/(8π), three sectors give 9/(8π), matching line 36. Lane K: CERTIFIED.", "evidence": [ "ComparatorChallenges/GaussianPropeller.lean:12: noncomputable def gaussian (d : ℕ) : Measure (Space d) := stdGaussian (Space d)", "ComparatorChallenges/GaussianPropeller.lean:14: def IsPartition {d k : ℕ} (A : Fin k → Set (Space d)) : Prop :=", "ComparatorChallenges/GaussianPropeller.lean:15: (∀ i, MeasurableSet (A i)) ∧ ∀ᵐ x ∂gaussian d, ∃! i, x ∈ A i", "ComparatorChallenges/GaussianPropeller.lean:16: ", "ComparatorChallenges/GaussianPropeller.lean:17: noncomputable def centroid {d : ℕ} (A : Set (Space d)) : Space d :=", "ComparatorChallenges/GaussianPropeller.lean:18: ∫ x in A, x ∂gaussian d", "ComparatorChallenges/GaussianPropeller.lean:33: (∀ (d k : ℕ), 0 < d → 0 < k → ∀ A : Fin k → Set (Space d),", "ComparatorChallenges/GaussianPropeller.lean:34: IsPartition A → value A ≤ 9 / (8 * Real.pi)) ∧", "ComparatorChallenges/GaussianPropeller.lean:35: (∀ (d k : ℕ), 2 ≤ d → 3 ≤ k →", "ComparatorChallenges/GaussianPropeller.lean:36: IsPartition (propeller d k) ∧ value (propeller d k) = 9 / (8 * Real.pi))" ] } ], "readings": [ "first/batch8.json", "flags/flags3.json" ] }, { "family": "097", "title": "The Euclidean Steinitz–Bergström bound", "headline": "Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within $`C\\sqrt d`$, independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order $`S_2(d)=\\Theta(\\sqrt d)`$.", "challenges": [ "SteinitzBergstrom" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within $`C\\sqrt d`$, independently of length.", "status": "stated", "coveredBy": [ "SteinitzBergstrom" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/SteinitzBergstrom.lean:24: theorem main : ∃ C : ℝ, SignedPrefixBound C ∧ OrderingPrefixBound C := by" ] }, { "clause": "Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums.", "status": "stated", "coveredBy": [ "SteinitzBergstrom" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/SteinitzBergstrom.lean:19: ∀ (v : Fin N → Vector d), (∀ i, ‖v i‖ ≤ 1) → (∑ i, v i) = 0 →" ] }, { "clause": "A matching lower bound gives the optimal order $`S_2(d)=\\Theta(\\sqrt d)`$.", "status": "undeclared", "coveredBy": [], "manuscript": "The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026", "evidence": [ "preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/consequences.tex:52: For the lower bounds, let $1\\le p\\le2$. The ordered standard basis", "preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/main.tex:46: $S_2(d)=\\Theta(\\sqrt d)$, and resolves the Euclidean", "lean/docs/097.md:9: The formalized result gives the Euclidean Steinitz–Bergström bound with one absolute constant $C$.", "lean/docs/097.md:9: The result is existential and does not supply an online algorithm." ] } ], "gap": "the matching lower bound S_2(d) = Ω(sqrt d), hence the Θ statement", "scopeNoteOnGap": "silent — docs/097.md:9 describes only the upper bounds (\"The formalized result gives the Euclidean Steinitz–Bergström bound with one absolute constant $C$. ... When the vector sum is zero, a permutation makes every unsigned prefix satisfy the same bound.\") and never mentions the lower bound", "note": "Both upper bounds are stated with one absolute C quantified before d and N. The Θ clause needs the lower bound S_2(d) ≥ c√d, which no challenge states; it comes from the one linked manuscript (an elementary few-line example, consequences.tex:52-68), and the note describes only the upper bounds without naming the omission. The gap is formal rather than substantive. Agrees with the first reader.", "firstSuspicious": "The omitted clause is a classical elementary lower bound (a few lines in the manuscript), so the gap is formal rather than substantive; the word follows the strict rule because the headline claims Θ and the scope note does not say the lower bound is omitted.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch8.json", "second/second3.json" ] }, { "family": "098", "title": "Compact counterexamples to bi-Lipschitz dimension reduction", "headline": "Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space. The doubling constant is universal. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.", "challenges": [ "DoublingHilbert", "CompactBanach" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space", "coveredBy": [ "CompactBanach" ], "note": "For every real Banach space B with B not finite-dimensional, there is a compact K, doubling with constant Lambda, that has no bi-Lipschitz embedding into any finite-dimensional real normed space E." }, { "clause": "The doubling constant is universal", "coveredBy": [ "CompactBanach" ], "note": "Lambda is chosen before B (there exists Lambda, for all B)." }, { "clause": "This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space", "coveredBy": [ "CompactBanach", "DoublingHilbert" ], "note": "CompactBanach applied to l2 (B is universe-polymorphic) gives a compact subset of Hilbert space. DoublingHilbert separately gives a doubling subset of l2 with constant 76800 and no embedding into any Euclidean R^k; it does not assert compactness." } ], "gap": null, "scopeNoteOnGap": "no gap", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "099", "title": "The sharp exponential scale of edit-distance distortion", "headline": "Determines the least distortion of embedding edit distance on words of length at most d into real ℓ1: it is $`\\exp(\\Theta(\\sqrt{\\log d\\,\\log\\log d}))`$. Insertions, deletions and substitutions have unit cost. The constants are uniform over all finite alphabets with at least two symbols, even when the alphabet grows with d; binary words already force the lower bound.", "challenges": [ "EditDistance", "FiniteCircle", "BinaryEditLower", "TreeEdit" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Determines the least distortion of embedding edit distance on words of length at most d into real ℓ1: it is $`\\exp(\\Theta(\\sqrt{\\log d\\,\\log\\log d}))`$.", "coveredBy": [ "EditDistance", "FiniteCircle" ], "note": "∃ c C > 0 ∃ d₀ ∀ d ≥ d₀: exp(c√(log d·log log d)) ≤ c₁(words of length ≤ d) ≤ exp(C√(log d·log log d))." }, { "clause": "Insertions, deletions and substitutions have unit cost.", "coveredBy": [ "EditDistance", "FiniteCircle", "BinaryEditLower", "TreeEdit" ], "note": "edit distance = least length of a script of unit insert/delete/substitute steps." }, { "clause": "The constants are uniform over all finite alphabets with at least two symbols, even when the alphabet grows with d", "coveredBy": [ "EditDistance", "FiniteCircle" ], "note": "c, C, d₀ chosen before the alphabet; alphabetSup over all finite alphabet sizes also bounded both ways." }, { "clause": "binary words already force the lower bound", "coveredBy": [ "EditDistance", "FiniteCircle", "BinaryEditLower", "TreeEdit" ], "note": "finite sets of ≥ 2 binary words of one common length n ≤ d with c₁ ≥ exp(c√(log d log log d))." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "100", "title": "Cylinder coverings below the half-area bound", "headline": "Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three.", "challenges": [ "CylinderCovering", "TriangularCovering", "RuledCovering" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area", "coveredBy": [ "TriangularCovering", "CylinderCovering" ], "note": "K is the closed regular tetrahedron of edge 2 (vertices (±1,0,0), (0,±1,√2); every cross-section is the rectangle |x| ≤ 1−t, |y| ≤ t). TriangularCovering.main asserts A_min = √2 (the true minimum projection area, which also pins the normalisation of Measure.euclideanHausdorffMeasure 2) and, for every small ε, a cover by 2⌈2/ε²⌉ TriCylinders (base = convex hull of 3 affinely independent points, perpendicular to a unit axis, compact) with totalArea < A_min/2. CylinderCovering's ExplicitAngularStatement repeats this for the explicit angular family." }, { "clause": "This disproves Bang's half-area cylinder-covering bound", "coveredBy": [ "TriangularCovering", "CylinderCovering" ], "note": "Same conjunct; additionally CylinderCovering states for EVERY regular tetrahedron (IsRegularTetrahedron) covers with parallelogram bases and with open bounded bases below half the minimum shadow area." }, { "clause": "the stronger directionwise normalized conjecture in dimension three", "coveredBy": [ "TriangularCovering", "CylinderCovering" ], "note": "NormalizedCounterexample: ∃ cover with Σ area(base_i)/shadowArea(axis_i) < 1/2; also relativeCost < 1/2 in ExplicitAngularStatement and in the every-regular-tetrahedron parallelogram/open statements. shadowArea of the axis is positive (K has interior, its projection is compact), so the division is not a junk /0." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": "Not headline gaps, but companion-paper discrepancies for a second reader: (a) the radial-sweep abstract and scope note say 'any tagged partition' / 'For every tagged partition', while CylinderCovering's RadialRiemann.cell only builds uniform N-cell partitions (tags arbitrary); (b) the ruled-sets abstract's 'fields that pass between the two regimes' (hyperbolic at some points, square-zero at others) is not stated — HyperbolicStatement needs 0 < Lλ at every point of D and the nilpotent statements need square-zero at every point (the scope note says 'The selected results cover …'); (c) RegularTriangularCounterexampleStatement is defined but not asserted; (d) the abstracts' affine extension to every nondegenerate tetrahedron is declared out (docs/100.md:22).", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "102", "title": "The Unique Games Conjecture and optimal approximation thresholds", "headline": "Proves Khot's Unique Games Conjecture. Independent direct reductions also establish NP-hardness, on unweighted graphs, of approximation beyond the Goemans–Williamson ratio for Max-Cut, below factor two for Vertex Cover, and within any fixed constant factor for Min-UnCut and directed feedback vertex set. These direct proofs use established PCP and Label Cover hardness results.", "challenges": [ "UniqueGamesTheorem", "OptimalMaxCut", "VertexCover", "MinUncut", "DirectedFeedback" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves Khot's Unique Games Conjecture", "coveredBy": [ "UniqueGamesTheorem" ], "note": "For every eps, delta in (0, 1/2): a polynomial-time reduction from 3SAT to unique games over a fixed alphabet with completeness >= 1 - eps and soundness <= delta." }, { "clause": "Independent direct reductions also establish NP-hardness, on unweighted graphs, of approximation beyond the Goemans–Williamson ratio for Max-Cut", "coveredBy": [ "OptimalMaxCut" ], "note": "For every alpha in (alpha_GW, 1]: a 3SAT gap reduction to simple unweighted graphs, with noBound < alpha * yesBound." }, { "clause": "below factor two for Vertex Cover", "coveredBy": [ "VertexCover" ], "note": "A gap of 1/2 + 1/m vs 1 - 1/m for every m >= 4, plus: any polynomial-time alpha-approximation with 1 <= alpha < 2 yields a polynomial-time 3SAT decider." }, { "clause": "and within any fixed constant factor for Min-UnCut and directed feedback vertex set", "coveredBy": [ "MinUncut", "DirectedFeedback" ], "note": "Min-UnCut: one machine for all K >= 2, polynomial for each fixed K, with opt <= threshold vs opt > K * threshold. DFVS: for every A >= 1 and every NP language, a gap reduction with dfvs <= k vs dfvs > A * k." }, { "clause": "These direct proofs use established PCP and Label Cover hardness results", "coveredBy": [], "note": "A statement about method, not a mathematical claim. Note that permitted_axioms are only propext, Quot.sound and Classical.choice, so no PCP or Label Cover result is assumed as an axiom." } ], "gap": null, "scopeNoteOnGap": "no gap", "note": null, "firstSuspicious": "The complexity model is Mathlib's TM2 multi-stack machine. A free-computation loophole would need an infinite stack alphabet, which the FiniteAlphabet field closes. A lane-K or second reader may still want to confirm that TM2 step semantics with finite state and alphabet types charge each step O(1).", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "103", "title": "Exact derandomization of logarithmic space: $`\\mathsf L=\\mathsf{RL}=\\mathsf{BPL}`$", "headline": "Proves $`\\mathsf L=\\mathsf{RL}=\\mathsf{BPL}`$, resolving derandomization for bounded-error logarithmic-space computation. An effective compiler converts each randomized polynomial-time logarithmic-space machine deciding a language with one-sided or two-sided error into a deterministic logarithmic-space decider with explicit polynomial running-time bounds.", "challenges": [ "LogspaceEquality" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves $`\\mathsf L=\\mathsf{RL}=\\mathsf{BPL}`$, resolving derandomization for bounded-error logarithmic-space computation", "coveredBy": [ "LogspaceEquality" ], "note": "L = RL ∧ RL = BPL for languages ⊆ List Bool, in an explicit multi-head two-way-input, multi-work-tape (ℤ-indexed, Bool alphabet) machine model with one fresh coin per step; RL/BPL require a polynomial clock c(n+2)^k by which every coin path has halted, one-sided 1/2 / two-sided 2/3–1/3 error, and log-space on every coin path at every time. AS WRITTEN this is the standard (polynomial-time) RL/BPL; BUT L, RL, BPL and 17 other definitions are definition holes (see flag)." }, { "clause": "An effective compiler converts each randomized polynomial-time logarithmic-space machine deciding a language with one-sided or two-sided error into a deterministic logarithmic-space decider with explicit polynomial running-time bounds", "coveredBy": [], "note": "Not stated: the Lean theorem is a classical equality of classes (∃ a deterministic machine, no effectivity, no explicit running-time bound). Declared in the scope note." } ], "gap": "The effective compiler and its explicit polynomial running-time bounds (headline's second sentence) are not stated; only the class equality is.", "scopeNoteOnGap": "docs/103.md:11: The selected theorem asserts equality of the language classes. The paper's explicit compiler and numerical running-time bounds are not separately stated in this Comparator target.", "note": null, "firstSuspicious": "The definition hole covers L, RL and BPL themselves: the statement read here is only as strong as lane K's check that the solution's bodies equal the challenge's. As written, the model is standard; I found no degenerate definition (LogSpace uses ⌈log₂(n+2)⌉ ≥ 1, coins are one-way, the clock is uniform over coin paths).", "confidence": "high", "conditionalOnLaneK": [ { "challenge": "LogspaceEquality", "strict": "SAME BODIES" } ], "flags": [ { "challenge": "LogspaceEquality", "flag": "DEFINITION HOLE: definition_names lists 20 declarations — Word, Language, CoinTape, Direction.move, Direction.moveInput, readInput, Machine.initial/step/run/spaceThrough/LogSpace/Deterministic/HaltsBy/Decides/extendCoins/acceptanceProbability, polynomialClock, and L, RL, BPL themselves. The theorem's entire content lives in these bodies: with L, RL, BPL as holes whose type (Set Language) is all Comparator compares, any bodies making the three sets equal would pass. The reading above is of the challenge's bodies; lane K must decide that the solution's bodies are these." } ], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "104", "title": "Quasipolynomial algorithms for mean-payoff, stochastic and parity games", "headline": "Gives deterministic algorithms using $`2^{O((\\log(L+2))^2)}`$ bit operations, for complete binary input length L, for ordinary mean-payoff games and two separate extensions. They compute exact values and optimal positional strategies in ordinary games, the nonnegative expectation-of-liminf value set in turn-based stochastic games, and the winning set for nonnegative liminf mean payoff conjoined with parity. Signed rewards, rational chance probabilities, and parity priorities are unrestricted and binary-encoded.", "challenges": [ "TruffetCounterexample", "RandomizedMeanPayoff" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Gives deterministic algorithms using $`2^{O((\\log(L+2))^2)}`$ bit operations, for complete binary input length L, for ordinary mean-payoff games", "status": "undeclared", "coveredBy": [], "manuscript": "Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026", "evidence": [ "CONTENTS.md:2597: We give a deterministic algorithm that computes the complete zero-threshold winning set of a finite mean-payoff game with arbitrary signed integer edge weights encoded in binary.", "lean/docs/104.md:5: - [Deterministic quasipolynomial-time mean-payoff games](../../preprints/Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf)", "lean/docs/104.md:12: The formalized result gives one randomized algorithm for the complete zero-threshold winning set of every finite mean-payoff game with signed binary weights.", "lean/ComparatorChallenges/RandomizedMeanPayoff.lean:118: 7 * 2 ^ T ≤ 8 * (Finset.univ.filter" ], "why": "The statement is randomized (correct on ≥ 7/8 of random tapes); a deterministic algorithm implies it, not conversely. The note links the deterministic paper, describes only the randomized result and the Truffet counterexample, and never says the deterministic algorithm is left out." }, { "clause": "and two separate extensions", "status": "unlinked", "coveredBy": [], "manuscript": "Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026; Mean-payoff-parity-games-in-quasipolynomial-time-October-5-2026", "evidence": [ "CONTENTS.md:2579: We give a uniform deterministic quasipolynomial-time algorithm for finite turn-based stochastic mean-payoff games", "CONTENTS.md:2588: We give a uniform deterministic quasipolynomial-time algorithm for mean-payoff parity games.", "lean/docs/104.md:6: - [Randomized quasipolynomial-time mean-payoff games](../../preprints/Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf)" ] }, { "clause": "They compute exact values and optimal positional strategies in ordinary games", "status": "undeclared", "coveredBy": [], "manuscript": "Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026", "evidence": [ "CONTENTS.md:2597: A reduction also computes the exact rational value at every vertex and globally optimal positional strategies for both players within the same quasipolynomial bound.", "lean/ComparatorChallenges/RandomizedMeanPayoff.lean:109: (∀ v : Fin G.n, c.tape 2 (v.val : ℤ) = some true ↔ Winning G v) ∧", "lean/docs/104.md:12: The formalized result gives one randomized algorithm for the complete zero-threshold winning set of every finite mean-payoff game with signed binary weights." ], "why": "The machine's output is the zero-threshold winning set only; no value and no strategy is output. The note is silent on values and strategies." }, { "clause": "the nonnegative expectation-of-liminf value set in turn-based stochastic games", "status": "unlinked", "coveredBy": [], "manuscript": "Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026", "evidence": [ "CONTENTS.md:2579: It computes exactly the vertices of nonnegative value, including value zero, for the expectation of the pathwise liminf mean payoff." ] }, { "clause": "and the winning set for nonnegative liminf mean payoff conjoined with parity", "status": "unlinked", "coveredBy": [], "manuscript": "Mean-payoff-parity-games-in-quasipolynomial-time-October-5-2026", "evidence": [ "CONTENTS.md:2588: It computes all vertices from which a player can enforce both nonnegative liminf mean payoff and the parity condition" ] }, { "clause": "Signed rewards", "part": "signed rewards, unrestricted and binary-encoded (ordinary games)", "status": "stated", "coveredBy": [ "RandomizedMeanPayoff" ], "evidence": [ "lean/ComparatorChallenges/RandomizedMeanPayoff.lean:47: decide (z < 0) :: encodeNat z.natAbs" ], "encoding": [ "stated for the randomized algorithm's input encoding (Game.weight : Fin m → ℤ, encodeInt)" ] }, { "clause": "rational chance probabilities, and parity priorities are unrestricted and binary-encoded.", "status": "unlinked", "coveredBy": [], "manuscript": "Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026; Mean-payoff-parity-games-in-quasipolynomial-time-October-5-2026", "evidence": [ "CONTENTS.md:2579: with signed integer rewards and rational chance-transition probabilities encoded in binary", "CONTENTS.md:2588: with arbitrary signed binary rewards and unrestricted binary priorities" ] } ], "gap": "Headline: deterministic quasipolynomial algorithms that compute values and optimal positional strategies (ordinary games), the nonnegative-value set of stochastic games, and mean-payoff-parity winning sets. Lean: one randomized (≥ 7/8 success) quasipolynomial algorithm for the zero-threshold winning set of ordinary games, plus a finite counterexample to Truffet's procedure. Of the four manuscripts, only the randomized one is formalized.", "scopeNoteOnGap": "silent on what is left out. It describes only what is formalized: lean/docs/104.md:12: \"The formalized result gives one randomized algorithm for the complete zero-threshold winning set of every finite mean-payoff game with signed binary weights. ...\" The note links two papers (lines 5–6); the stochastic and parity manuscripts are not linked, and determinism, values and strategies are not mentioned.", "note": "No headline clause is stated in full, but the headline implies RandomizedMeanPayoff (a deterministic quasipolynomial algorithm for values gives the winning set with probability 1, up to a polynomial machine-model translation), so NARROWER, not DIFFERENT. The stochastic and parity clauses are UNLINKED; the deterministic algorithm and the values/positional strategies come from a paper the note links, and the note only describes the randomized result, so NARROWER — UNDECLARED. Agree with the first reader. TruffetCounterexample covers no headline clause.", "firstSuspicious": "Two scope-note conventions are possible. This reading counts as DECLARED only an omission the note names. A reader who counts an accurate description of the formalized content as a declaration would call this NARROWER — DECLARED. The family's main headline object (deterministic, values/strategies, stochastic, parity) is entirely outside the Lean.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "TruffetCounterexample", "flag": "DEFINITION carries the conclusion: the counterexample is only as meaningful as the encoded procedure (TruffetCounterexample.lean:39-91); a second reader comparing it with Truffet's text is needed if this row is cited." } ], "flagReadings": null, "readings": [ "first/batch5.json", "second/second4.json" ] }, { "family": "105", "title": "Perfect completeness for 2-to-1 games", "headline": "Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational $`\\delta\\in(0,1)`$, it is NP-hard to distinguish satisfiable games from games whose optimum is at most δ, on explicit unweighted instances. The alphabet depends only on δ, and every right-hand label has exactly two preimages under each constraint map.", "challenges": [ "PerfectCompleteness" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational $`\\delta\\in(0,1)`$, it is NP-hard to distinguish satisfiable games from games whose optimum is at most δ, on explicit unweighted instances.", "coveredBy": [ "PerfectCompleteness" ], "note": "a polynomial-time TM2 many-one reduction from binary-encoded 3SAT to 2-to-1 games with value 1 on YES and <= δ on NO; NP-hardness follows from NP-hardness of 3SAT (Cook–Levin), which is not itself stated — the standard encoding of NP-hardness" }, { "clause": "The alphabet depends only on δ, and every right-hand label has exactly two preimages under each constraint map.", "coveredBy": [ "PerfectCompleteness" ], "note": "alphabet is a field of the reduction structure, fixed before the input; ProjectionTable.exactlyTwo" } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "PerfectCompleteness", "flag": "JUNK? cleared: Instance.nonempty forbids an empty edge list, so value never divides by zero" } ], "flagReadings": [ { "challenge": "PerfectCompleteness", "flag": "JUNK? cleared: Instance.nonempty forbids an empty edge list, so value never divides by zero", "decision": "cleared", "why": "value = maxSatisfied / edges.length, and Instance.nonempty (edges ≠ []) makes the length ≥ 1. So value is the true fraction in [0,1], never 0/0 = 0. That junk would have satisfied soundness for free; completeness (value = 1) would already have excluded it on satisfiable inputs. maxSatisfied is a Finset.sup over the finite labelling type, a true maximum. Nonempty edges also force leftVertices, rightVertices ≥ 1, and alphabet ≥ 2.", "evidence": [ "ComparatorChallenges/PerfectCompleteness.lean:112-116: structure Instance (q : Nat) where ⏎ leftVertices : Nat ⏎ rightVertices : Nat ⏎ edges : List (Edge leftVertices rightVertices q) ⏎ nonempty : edges ≠ []", "ComparatorChallenges/PerfectCompleteness.lean:134-138: def maxSatisfied {q : Nat} (G : Instance q) : Nat := ⏎ Finset.univ.sup (fun s : Labeling G => countSatisfied s.1 s.2 G.edges) ⏎ ⏎ noncomputable def value {q : Nat} (G : Instance q) : ℝ := ⏎ (G.maxSatisfied : ℝ) / G.edges.length", "ComparatorChallenges/PerfectCompleteness.lean:165-173: structure BinaryGapReduction (δ : ℚ) where ⏎ alphabet : Nat ⏎ alphabet_ge_two : 2 ≤ alphabet ⏎ construct : List Bool → Instance alphabet ⏎ computation : TM2ComputableInPolyTime (id : List Bool → List Bool) ⏎ Encoding.gameBits construct ⏎ finiteAlphabet : MachineFiniteAlphabet.FiniteAlphabet computation.tm ⏎ completeness : ∀ input, BinaryLanguage.language input → (construct input).value = 1 ⏎ soundness : ∀ input, ¬BinaryLanguage.language input → (construct input).value ≤ (δ : ℝ)" ] } ], "readings": [ "first/batch8.json", "flags/flags4.json" ] }, { "family": "106", "title": "Hardness of coloring three-colorable graphs", "headline": "It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed $`0\\lt \\delta\\lt 1/3`$, a deterministic polynomial-time reduction from 3SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size $`\\delta n`$ otherwise, where n is the number of vertices.", "challenges": [ "IndependentSets" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors.", "coveredBy": [ "IndependentSets" ], "note": "Implied, not stated: take δ = 1/(c+1) < 1/3; in the unsatisfiable case α(G) < n/(c+1) forces χ(G) > c, so a polynomial-time c-colourer for 3-colourable graphs would decide 3SAT through the reduction. 'NP-hard' itself (Cook–Levin for 3SAT) is outside Lean." }, { "clause": "More strongly, for every fixed $`0\\lt \\delta\\lt 1/3`$, a deterministic polynomial-time reduction from 3SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size $`\\delta n`$ otherwise, where n is the number of vertices.", "coveredBy": [ "IndependentSets" ], "note": "Stated directly (∀ δ ∈ (0,1/3), Nonempty (RealGraphReduction δ))." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch1.json" ] }, { "family": "107", "title": "Matrix multiplication with exponent at most 9/4", "headline": "Proves $`\\omega\\le9/4`$ over ℂ, giving $`O_\\varepsilon(n^{9/4+\\varepsilon})`$ arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension na with a > 0.465 permits $`n^{2+o(1)}`$ rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field.", "challenges": [ "MatrixMultiplication", "MatrixFields" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Proves $`\\omega\\le9/4`$ over ℂ, giving $`O_\\varepsilon(n^{9/4+\\varepsilon})`$ arithmetic operations for square matrix multiplication.", "status": "stated", "coveredBy": [ "MatrixMultiplication" ], "reading": "complex_omega_le_nine_quarters: ω(ℂ) ≤ 9/4, ω the sInf of exponents τ admitting, for every ε > 0, correct division-free programs of cost ≤ C·n^{τ+ε}; the O_ε(n^{9/4+ε}) form follows from the definition. Not junk: the admissible set contains 3 and every program costs ≥ n² (each output entry needs its own gate), so the sInf is a true infimum.", "evidence": [ "lean/ComparatorChallenges/MatrixMultiplication.lean:95: def omega (F : Type*) [Field F] : ℝ := sInf {τ : ℝ | AdmissibleExponent F τ}", "lean/ComparatorChallenges/MatrixMultiplication.lean:118: theorem complex_omega_le_nine_quarters :", "lean/ComparatorChallenges/MatrixMultiplication.lean:119: Arithmetic.omega ℂ ≤ (9 : ℝ) / 4 := by" ] }, { "clause": "In characteristic zero, some inner dimension na with a > 0.465 permits $`n^{2+o(1)}`$ rectangular multiplication.", "part": "fields of characteristic zero other than ℂ (ℂ itself is stated by complex_alpha_gt_93_div_200)", "status": "undeclared", "coveredBy": [ "MatrixMultiplication" ], "manuscript": "Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026", "classicalTheorem": "Schönhage (1981): the matrix-multiplication exponents of a field depend only on its characteristic — not formalized; only through it does the ℂ statement give every characteristic-zero field.", "evidence": [ "lean/ComparatorChallenges/MatrixMultiplication.lean:110: sSup {k : ℝ | k ∈ Set.Icc 0 1 ∧ rectangularOmega ℂ k = 2}", "lean/ComparatorChallenges/MatrixMultiplication.lean:123: (93 : ℝ) / 200 < Arithmetic.complexAlpha := by", "CONTENTS.md:2663: Over every field of characteristic zero, we prove that the square matrix-multiplication exponent satisfies ω < 2.258, the dual exponent satisfies α > 0.465", "lean/docs/107.md:5: - [Complex Matrix Multiplication Below 2.258 and Rectangular Bounds](../../preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026.pdf)", "lean/docs/107.md:10: The formalized results bound the complex matrix-multiplication exponent by $\\omega(\\mathbb C)\\le9/4$, the dual exponent by $\\alpha>0.465$" ] }, { "clause": "Further square bounds give ω < 2.258 outside finitely many positive characteristics", "status": "undeclared", "coveredBy": [], "manuscript": "Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026", "evidence": [ "lean/ComparatorChallenges/MatrixFields.lean:110: Arithmetic.omega F < (2371054886006746 : ℝ) / 10 ^ 15 ∧", "CONTENTS.md:2663: The strict square and k = 0.709 rectangular bounds also hold over every field except possibly in one finite set of positive characteristics, in the arithmetic-operation model.", "lean/docs/107.md:5: - [Complex Matrix Multiplication Below 2.258 and Rectangular Bounds](../../preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026.pdf)", "lean/docs/107.md:10: The square bound implies the paper's weaker $2.258$ headline bound.", "lean/docs/107.md:12: The formalized result gives the unconditional bound $\\omega(F)<2.371054886006746$ for every field $F$, including finite fields and fields of positive characteristic." ] }, { "clause": "and ω < 2.371054886006746 over every fixed field.", "status": "stated", "coveredBy": [ "MatrixFields" ], "reading": "omega_lt_source_constant: ω(F) < 2371054886006746/10^15 for every field F (any universe), same program model.", "evidence": [ "lean/ComparatorChallenges/MatrixFields.lean:109: theorem omega_lt_source_constant (F : Type u) [Field F] :", "lean/ComparatorChallenges/MatrixFields.lean:110: Arithmetic.omega F < (2371054886006746 : ℝ) / 10 ^ 15 ∧" ] } ], "gap": "'ω < 2.258 outside finitely many positive characteristics' is not stated for any field of positive characteristic; the characteristic-zero rectangular clause is stated over C only. Both would follow from the C results by standard characteristic-transfer arguments (Schönhage-type), but no challenge states them.", "scopeNoteOnGap": "silent on positive characteristics; the note's only 2.258 sentence is lean/docs/107.md:10: The square bound implies the paper's weaker $2.258$ headline bound. — and it names C for the other bounds: lean/docs/107.md:10: The formalized results bound the complex matrix-multiplication exponent by $\\omega(\\mathbb C)\\le9/4$, the dual exponent by $\\alpha>0.465$", "note": "ω(ℂ) ≤ 9/4, α(ℂ) > 0.465 and the every-field 2.371 bound are stated. Not stated: ω < 2.258 for fields of positive characteristic outside a finite set (and for characteristic-zero fields other than ℂ), and α > 0.465 beyond ℂ; both come from the linked 'Below 2.258' manuscript, and docs/107.md describes only the formalized ℂ and 2.371 bounds without naming either omission. Agrees with the first reader, who read 'In characteristic zero' → ℂ as declared by the note's 'complex'; I read it as undeclared too (the note's 'complex' qualifies ω, and the transfer needs Schönhage's theorem), which does not change the family word.", "firstSuspicious": "Second reader: (1) whether 'In characteristic zero' → C-only counts as a named encoding (I treat it as declared by the scope note's 'complex'); the undeclared part is the positive-characteristic 2.258 clause. (2) The scope note's paper list omits the October 2 'Nine-Fourths' paper although its 9/4 bound is what is stated.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json", "second/second5.json" ] }, { "family": "108", "title": "A cubic permanent–determinant lower bound", "headline": "Proves an $`\\Omega(n^3)`$ lower bound for the border determinantal complexity of the $`n\\times n`$ permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least $`cn^3`$, for an absolute c > 0 and all sufficiently large n; the same bound therefore holds for exact representations.", "challenges": [ "PermanentCubic", "SmoothInitialForm" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves an $`\\Omega(n^3)`$ lower bound for the border determinantal complexity of the $`n\\times n`$ permanent over ℂ", "coveredBy": [ "PermanentCubic" ], "note": "m >= 1408 and HasBorderDeterminant n (perm_m) imply n >= m^3/(5529600 e)." }, { "clause": "Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least $`cn^3`$, for an absolute c > 0 and all sufficiently large n", "coveredBy": [ "PermanentCubic" ], "note": "Explicit c = 1/(5529600 e) and explicit threshold m >= 1408. The border notion is: every coefficient of det(A0(j) + sum_v A(j)_v X_v) converges to the corresponding coefficient of the permanent." }, { "clause": "the same bound therefore holds for exact representations", "coveredBy": [ "PermanentCubic" ], "note": "Second conjunct (HasExactDeterminant)." } ], "gap": null, "scopeNoteOnGap": "no gap (docs/108.md:11 places the algebraic-branching-program consequences outside; they are not in the headline)", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "110", "title": "Optimal-order randomized k-server on arbitrary metrics", "headline": "Establishes a randomized competitive ratio $`O(\\log^2(k+1))`$ for k-server on every metric space, matching the worst-case lower-bound order. One policy serves every finite oblivious request sequence, including on infinite unbounded metrics. On finite rational metrics, a uniform implementation has polynomial preprocessing and per-request bit cost in the input length and $`\\log(t+1)`$ at request t, with a finite instance-dependent additive movement constant.", "challenges": [ "KServer", "UniformKServer" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Establishes a randomized competitive ratio $`O(\\log^2(k+1))`$ for k-server on every metric space, matching the worst-case lower-bound order.", "coveredBy": [ "KServer" ], "note": "∃ C>0 ∀ k>=2 ∀ metric X with >= k+1 points; trivial cases (k=1, <= k points) excluded as in the scope note. 'matching the worst-case lower-bound order' cites a known lower bound and is not a claim of this family. CAUTION: every definition in KServer is a Comparator definition hole (see flags)." }, { "clause": "One policy serves every finite oblivious request sequence, including on infinite unbounded metrics.", "coveredBy": [ "KServer" ], "note": "∀ initial configuration s ∃ policy A ∃ B ∀ request lists; X is an arbitrary metric space in an arbitrary universe" }, { "clause": "On finite rational metrics, a uniform implementation has polynomial preprocessing and per-request bit cost in the input length and $`\\log(t+1)`$ at request t, with a finite instance-dependent additive movement constant.", "coveredBy": [ "UniformKServer" ], "note": "one finite-control multi-tape randomized bit machine M with constants c,e,C for all instances with 3<=n, 2<=k if M.Indep (I : Set (Fin n)) then ∑ e ∈ I, v e else 0)", "lean/ComparatorChallenges/MatroidProphet.lean:83: Integrable (fun ω => optimum M (V ω)) μ →", "lean/ComparatorChallenges/MatroidProphet.lean:85: Integrable (fun ω => reward A (R ω) (S ω) (V ω) (π ω)) μ ∧", "lean/ComparatorChallenges/MatroidProphet.lean:86: ((2 : ℝ) ^ 310)⁻¹ * (∫ ω, optimum M (V ω) ∂μ) ≤", "lean/ComparatorChallenges/MatroidProphet.lean:87: ∫ ω, reward A (R ω) (S ω) (V ω) (π ω) ∂μ" ] } ], "readings": [ "first/batch3.json", "flags/flags2.json" ] }, { "family": "112", "title": "Beyond the square-root exponent for depth-three circuits", "headline": "Constructs a single language in deterministic polynomial time whose n-bit membership function requires $`2^{\\omega(\\sqrt n)}`$ total gates in unbounded-fan-in OR–AND–OR circuits, at every sufficiently large input length. This crosses the square-root-exponent threshold for explicit depth-three Boolean circuit lower bounds.", "challenges": [ "DepthThree" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a single language in deterministic polynomial time whose n-bit membership function requires $`2^{\\omega(\\sqrt n)}`$ total gates in unbounded-fan-in OR–AND–OR circuits, at every sufficiently large input length", "coveredBy": [ "DepthThree" ], "note": "There exist L, a finite multi-tape machine M and C, a > 0 with M deciding L in C(|w|+1)^a steps, such that for all A > 0 there is N with every OR-AND-OR circuit for L on n >= N bits having more than 2^(A sqrt n) gates." }, { "clause": "This crosses the square-root-exponent threshold for explicit depth-three Boolean circuit lower bounds", "coveredBy": [ "DepthThree" ], "note": "Same theorem." } ], "gap": null, "scopeNoteOnGap": "no gap", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "113", "title": "Approximate counting and entropy of perfect matchings", "headline": "Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points.", "challenges": [ "MatchingFPRAS", "MatchingEntropy", "BinaryMatching", "SingletonLoopMatching", "MatchingEntropyBounds", "TriangleFace" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts.", "coveredBy": [ "MatchingFPRAS" ], "note": "One fixed finite-alphabet randomized machine; on every tape it halts within C·(len + ⌈1/ε⌉ + clog₂⌈1/δ⌉ + 1)^d ticks with a nonnegative rational; a zero count gives output 0 on every tape; ≥ 1−δ of tapes give relative error ≤ ε." }, { "clause": "Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points.", "coveredBy": [ "MatchingEntropyBounds" ], "note": "For every loopless multigraph on 2m ≥ 2 vertices with a perfect matching and every x in the convex hull of perfect-matching indicators: F(x) − (2 − 2/m)B(x) ≤ H(x) ≤ F(x)." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "MatchingEntropyBounds", "flag": "JUNK? (checked, benign): H(x) is an sSup over ℝ; for x in the polytope the feasible set is nonempty and entropies are bounded by log #matchings, so sSup is the true maximum." } ], "flagReadings": [ { "challenge": "MatchingEntropyBounds", "flag": "JUNK? (checked, benign): H(x) is an sSup over ℝ; for x in the polytope the feasible set is nonempty and entropies are bounded by log #matchings, so sSup is the true maximum.", "decision": "cleared", "why": "maxMatchingEntropy is a Real sSup of `entropy '' feasibleLaws G x`. Nonempty: hx puts x in the convex hull of the indicator vectors of the finitely many perfect matchings (G.Matching is a Fintype), so some p in the probability simplex on G.Matching has mean p = x. Bounded above: Σ negMulLog p_M ≤ log(card G.Matching) for any probability vector. The feasible set is also compact, so the supremum is a genuine maximum entropy. Neither junk case (empty, unbounded) is reachable, so both inequalities compare against the true H(x). Lane K: not run yet.", "evidence": [ "ComparatorChallenges/MatchingEntropyBounds.lean:41: def polytope (G : LooplessGraph V E) : Set (E → ℝ) :=", "ComparatorChallenges/MatchingEntropyBounds.lean:42: convexHull ℝ (Set.range G.indicator)", "ComparatorChallenges/MatchingEntropyBounds.lean:52: def LooplessGraph.feasibleLaws (G : LooplessGraph V E) (y : E → ℝ) : Set (G.Matching → ℝ) :=", "ComparatorChallenges/MatchingEntropyBounds.lean:53: {p | p ∈ probabilitySimplex G.Matching ∧ G.mean p = y}", "ComparatorChallenges/MatchingEntropyBounds.lean:55: def entropy {ι : Type uι} [Fintype ι] (p : ι → ℝ) : ℝ :=", "ComparatorChallenges/MatchingEntropyBounds.lean:56: ∑ i, Real.negMulLog (p i)", "ComparatorChallenges/MatchingEntropyBounds.lean:57: ", "ComparatorChallenges/MatchingEntropyBounds.lean:58: def maxMatchingEntropy (G : LooplessGraph V E) (y : E → ℝ) : ℝ :=", "ComparatorChallenges/MatchingEntropyBounds.lean:59: sSup (entropy '' G.feasibleLaws y)", "ComparatorChallenges/MatchingEntropyBounds.lean:73: theorem Refined.pointwise_entropy (G : LooplessGraph V E) (m : ℕ)", "ComparatorChallenges/MatchingEntropyBounds.lean:74: (hm : 0 < m) (hV : Fintype.card V = 2 * m) (hPM : Nonempty G.Matching)", "ComparatorChallenges/MatchingEntropyBounds.lean:75: (x : E → ℝ) (hx : x ∈ G.polytope) :", "ComparatorChallenges/MatchingEntropyBounds.lean:76: entropy x - (2 - 2 / (m : ℝ)) * complementEntropy x ≤ maxMatchingEntropy G x ∧", "ComparatorChallenges/MatchingEntropyBounds.lean:77: maxMatchingEntropy G x ≤ entropy x := by" ] } ], "readings": [ "first/batch3.json", "flags/flags3.json" ] }, { "family": "114", "title": "Approximate counting of common integer polymatroid bases", "headline": "Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles. Capacities are binary-encoded, each integer vector counts once, and oracle calls and bit operations outside the oracles are polynomial on every execution. For matroids presented by independence oracles, the results also cover common independent sets of prescribed, unrestricted, or maximum cardinality, even when the ranks differ.", "challenges": [ "CommonBasesFPRAS" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles.", "status": "unlinked", "coveredBy": [], "manuscript": "An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026", "partial": "the 0/1-capacity special case (common bases of two equal-rank matroids, independence oracles) is stated by CommonBasesFPRAS", "evidence": [ "preprints/An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026/build/main.tex:64-66: We give a fully polynomial randomized approximation scheme for counting ⏎ common integer bases of two polymatroids with the same total rank, supplied ⏎ by exact rank-value oracles. The total rank and capacities are encoded in", "lean/docs/114.md:5: - [Approximate counting of common bases of two matroids](../../preprints/Approximate-counting-of-common-bases-of-two-matroids-September-23-2026/main.pdf)", "lean/ComparatorChallenges/CommonBasesFPRAS.lean:7-16: abbrev IndependenceOracle (n : ℕ) := Bool → (Fin n → Bool) → Bool ⏎ def ExactOracles {n : ℕ} (M₁ M₂ : Matroid (Fin n)) ⏎ (O : IndependenceOracle n) : Prop := ⏎ ∀ (which : Bool) (A : Fin n → Bool), ⏎ O which A = true ↔ (if which then M₂ else M₁).Indep {i | A i = true} ⏎ noncomputable def commonBaseCount {n : ℕ} (M₁ M₂ : Matroid (Fin n)) : ℕ := ⏎ Nat.card {A : Finset (Fin n) // M₁.IsBase (A : Set (Fin n)) ∧ ⏎ M₂.IsBase (A : Set (Fin n))}" ] }, { "clause": "Capacities are binary-encoded, each integer vector counts once, and oracle calls and bit operations outside the oracles are polynomial on every execution.", "status": "unlinked", "coveredBy": [], "manuscript": "An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026", "partial": "the per-execution polynomial bound on oracle calls and bit operations is stated for the matroid case (every random tape)", "evidence": [ "preprints/An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026/build/main.tex:66-71: by exact rank-value oracles. The total rank and capacities are encoded in ⏎ binary, and each integer vector is counted once. On every execution, the ⏎ number of oracle calls and the bit work outside the oracles are bounded ⏎ by a fixed polynomial in the ground-set size, the binary input length, ⏎ the inverse relative-error tolerance, and the logarithm of the inverse ⏎ failure probability. The algorithm handles binary capacities directly,", "lean/docs/114.md:5: - [Approximate counting of common bases of two matroids](../../preprints/Approximate-counting-of-common-bases-of-two-matroids-September-23-2026/main.pdf)", "lean/ComparatorChallenges/CommonBasesFPRAS.lean:117-120: (∀ bits : ℕ → Bool, ⏎ let out := execution P O (initialConfig k s n r ε δ) bits B ⏎ out.output.isSome ∧ out.oracleCalls ≤ B ∧ out.bitOperations ≤ B ∧ ⏎ ∀ z, out.output = some z → 0 ≤ z ∧ (Z = 0 → z = 0)) ∧" ] }, { "clause": "For matroids presented by independence oracles, the results also cover common independent sets of prescribed, unrestricted, or maximum cardinality, even when the ranks differ.", "status": "undeclared", "coveredBy": [], "manuscript": "Approximate-counting-of-common-bases-of-two-matroids-September-23-2026", "evidence": [ "preprints/Approximate-counting-of-common-bases-of-two-matroids-September-23-2026/build/main.tex:106-108: The theorem also yields an FPRAS for counting common independent sets ⏎ of prescribed, unrestricted, or maximum cardinality, even when the two ⏎ matroid ranks differ, and an almost-uniform sampler for each of these", "preprints/Approximate-counting-of-common-bases-of-two-matroids-September-23-2026/build/main.tex:1479-1483: \\begin{corollary}[Common independent sets]\\label{cor:intersection} ⏎ Let $M_1,M_2$ be arbitrary matroids on an explicitly enumerated ground ⏎ set $[n]$, supplied by fixed exact independence oracles with $n$-bit ⏎ subset queries. Their ranks may differ. For an integer $k$, let ⏎ $\\mathcal C_k$ consist of the $k$-element subsets of $[n]$ independent", "lean/ComparatorChallenges/CommonBasesFPRAS.lean:113-116: M₁.eRank = (r : ℕ∞) → M₂.eRank = (r : ℕ∞) → ⏎ ExactOracles M₁ M₂ O → 0 < ε → ε < 1 → 0 < δ → δ < 1 → ⏎ let B := resourceBound C d n r ε δ ⏎ let Z := commonBaseCount M₁ M₂", "lean/ComparatorChallenges/CommonBasesFPRAS.lean:14-16: noncomputable def commonBaseCount {n : ℕ} (M₁ M₂ : Matroid (Fin n)) : ℕ := ⏎ Nat.card {A : Finset (Fin n) // M₁.IsBase (A : Set (Fin n)) ∧ ⏎ M₂.IsBase (A : Set (Fin n))}", "lean/docs/114.md:9: The formalized result gives a fully polynomial randomized approximation scheme for the number of common bases of two equal-rank matroids on an enumerated finite ground set, using independence oracles. For rational $0<\\varepsilon,\\delta<1$, the nonnegative rational output has relative error at most $\\varepsilon$ with probability at least $1-\\delta$. A zero count produces zero on every execution. Oracle calls and bit operations have polynomial bounds on every random tape in the input size, $\\varepsilon^{-1}$, and $\\log(\\delta^{-1})$." ] } ], "gap": "The headline object (common integer bases of integral polymatroids with binary capacities and rank-value oracles) and the independent-set variants are not stated. Only the matroid common-bases FPRAS is stated (the September 23 paper; the October 5 polymatroid paper is not formalized).", "scopeNoteOnGap": "silent. lean/docs/114.md:9 describes only the matroid result (\"The formalized result gives a fully polynomial randomized approximation scheme for the number of common bases of two equal-rank matroids ... using independence oracles.\"). The note links only the matroid paper (line 5), and polymatroids are mentioned only in the title (line 1).", "note": "Only the matroid common-bases FPRAS of the linked September-23 paper is stated (equal ranks forced by M₁.eRank = M₂.eRank = r). The polymatroid headline is the unlinked October-5 paper; the independent-set corollary (prescribed/unrestricted/maximum cardinality, different ranks) is in the linked paper (main.tex:106-108, Corollary cor:intersection) and the note is silent on it, so the family stays UNDECLARED even with the UNLINKED split. Agree with the first reader.", "firstSuspicious": "Same scope-note convention issue as 104: the note scopes the formalized content accurately but never says the polymatroid result is left out.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "CommonBasesFPRAS", "flag": "JUNK? checked, none: the halt instruction divides two register values in ℚ (x/0 = 0), and a malformed query outputs 0; both only produce wrong answers when Z > 0, which the accuracy clause excludes." } ], "flagReadings": null, "readings": [ "first/batch5.json", "second/second6.json" ] }, { "family": "115", "title": "Sampling and counting contingency tables with arbitrary margins", "headline": "For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and binary-encoded margins are unrestricted. Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution.", "challenges": [ "ContingencyTables" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time.", "coveredBy": [ "ContingencyTables" ], "note": "exactSampling: halted outputs are feasible tables, P(output = X) → 1/|Ω| for every X, halting has probability → 1 and Σ_t P(T > t) ≤ C·size^d (expected running time). boundedSampling: every tape of the fixed polynomial length outputs a table and the TV distance to uniform is ≤ 2^(−k)." }, { "clause": "The dimensions and binary-encoded margins are unrestricted.", "coveredBy": [ "ContingencyTables" ], "note": "∀ m n, arbitrary margins with equal totals; the time bound is polynomial in m + n + log₂(total) + k (≤ input length), which is a stronger bound." }, { "clause": "Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution.", "coveredBy": [ "ContingencyTables" ], "note": "counting: for m, n > 0, any margins with equal totals, any cell bounds b (zero allowed), rational 0<ε,δ<1, every tape halts within a polynomial in input length, 1/ε, log 1/δ; count 0 ⇒ output 0 on every tape; ≥ 1−δ of tapes are ε-accurate." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "ContingencyTables", "flag": "ENCODING (not a gap): the counting statement quantifies only over 0 < m, 0 < n and equal totals (ContingencyTables.lean:196-198); the headline's 'arbitrary' margins with unequal totals have count 0 trivially." } ], "flagReadings": null, "readings": [ "first/batch3.json" ] }, { "family": "116", "title": "Uniform black-box noncommutative identity testing across characteristics", "headline": "For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that characteristic. Rational formulas over ℚ also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation.", "challenges": [ "FormulaHitting", "RationalHitting" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that characteristic.", "status": "stated", "coveredBy": [ "FormulaHitting" ], "evidence": [ "lean/ComparatorChallenges/FormulaHitting.lean:46: def dimension (n s : ℕ) : ℕ := if n = 1 ∧ s = 1 then 1 else 2 * n * s ^ 2", "lean/ComparatorChallenges/FormulaHitting.lean:57: theorem universal_hitting (n s : ℕ) (hn : 1 ≤ n) (hs : 1 ≤ s)", "lean/ComparatorChallenges/FormulaHitting.lean:58: (F : Type u_2) [Field F] [CharZero F] (f : Formula F n)", "lean/ComparatorChallenges/FormulaHitting.lean:59: (hsize : f.size ≤ s) (hf : ∃ w, f.coeff w ≠ 0) :", "lean/ComparatorChallenges/FormulaHitting.lean:60: f.eval (Matrix.scalar (Fin (dimension n s)))", "lean/ComparatorChallenges/FormulaHitting.lean:61: (fun i r q => (generator n s i r q : F)) ≠ 0 := by" ], "note": "Read for characteristic ZERO only: one explicit rational tuple of dimension 2ns² (1 when n = s = 1) detects every nonzero division-free formula of size ≤ s over every characteristic-zero field (size counts leaves and gates; nonzero = some word coefficient nonzero)." }, { "clause": "constructs in deterministic polynomial bit time", "status": "declared", "coveredBy": [], "manuscript": "One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026", "evidence": [ "lean/docs/116.md:12: The selected theorem states this universal hitting property. The paper's deterministic polynomial bit-construction bound and matrix-dimension bound $O(ns^2)$ are not separately asserted in it." ], "note": "Characteristic zero: the tuple is explicit, but no complexity statement is made for it; the note names the omission." }, { "clause": "For each characteristic", "status": "unlinked", "coveredBy": [], "manuscript": "Uniform-Matrix-Hitting-Points-in-Every-Positive-Characteristic-October-4-2026", "evidence": [ "CONTENTS.md:2854: [Uniform Matrix Hitting Points in Every Positive Characteristic]", "CONTENTS.md:2856: We construct a single matrix substitution that detects every nonzero size-s division-free noncommutative formula in n variables over every field of a given positive characteristic.", "lean/ComparatorChallenges/FormulaHitting.lean:58: [CharZero F]", "lean/docs/116.md:5: - [One Rational Matrix Hitting Point for Noncommutative Formulas](../../preprints/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026.pdf)", "lean/docs/116.md:6: - [Polynomial Hitting Lists for Noncommutative Rational Formulas](../../preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026.pdf)" ], "note": "Every positive characteristic (the whole first sentence read for char p, including its polynomial-time construction over 𝔽_p) is unstated; it comes from the October-4 manuscript the note does not link." }, { "clause": "Rational formulas over ℚ also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation.", "status": "stated", "coveredBy": [ "RationalHitting" ], "evidence": [ "lean/ComparatorChallenges/RationalHitting.lean:95: def Hits {n : ℕ} (s : ℕ) (H : Output n) : Prop :=", "lean/ComparatorChallenges/RationalHitting.lean:96: ∀ f : Formula n, f.size ≤ s → Admissible f → Nonzero f →", "lean/ComparatorChallenges/RationalHitting.lean:97: ∃ X ∈ H.tuples, ∃ v, Evaluates f X v ∧ IsUnit v", "lean/ComparatorChallenges/RationalHitting.lean:102: ∃ (m : ℕ) (G : Machine m) (C k : ℕ), 0 < C ∧ 0 < k ∧", "lean/ComparatorChallenges/RationalHitting.lean:106: OutputsWithin G s (C * (n + s + 1) ^ k) H ∧ Hits s H" ], "note": "One fixed TM0 machine on unary (n, s); dimension, binary output length and step count all ≤ C(n+s+1)^k; inverse gates evaluated relationally (a genuine two-sided inverse), so no junk inverse." } ], "gap": "Every positive characteristic: no challenge states a hitting tuple over 𝔽_p or over fields of characteristic p (the October-4 manuscript 'Uniform Matrix Hitting Points in Every Positive Characteristic', which the scope note does not link). Also not asserted, but declared: the polynomial bit-time construction of the characteristic-zero tuple.", "scopeNoteOnGap": "silent on positive characteristic; the only declared omission is: The selected theorem states this universal hitting property. The paper's deterministic polynomial bit-construction bound and matrix-dimension bound $O(ns^2)$ are not separately asserted in it.", "note": "Characteristic zero and the ℚ rational-formula lists are stated; the char-0 bit-time bound is declared (docs/116.md:12); every positive characteristic comes from the October-4 manuscript docs/116.md does not link (it links the two September-24 papers). Most severe clause: unlinked. The first reader found the same gap and called it UNDECLARED before amendment 3; the difference is the amendment.", "firstSuspicious": "The note links only the two characteristic-zero/ℚ manuscripts and never mentions positive characteristic.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "FormulaHitting", "flag": "NARROWER (undeclared): [CharZero F] — positive characteristic excluded" } ], "flagReadings": null, "readings": [ "first/batch1.json", "second/second7.json" ] }, { "family": "117", "title": "Uniform sparsest cut: hardness and semidefinite gaps", "headline": "Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands. The Goemans–Linial semidefinite relaxation also has integrality gaps of order at least $`\\sqrt{\\log n}/(\\log\\log n)^3`$, approaching the square-root-logarithmic upper bound.", "challenges": [ "UniformSparsestCut" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands.", "coveredBy": [], "note": "not stated; declared out ('not ... a computational hardness theorem'); its manuscript (Constant-factor-hardness-of-uniform-sparsest-cut) is not linked from the scope note" }, { "clause": "The Goemans–Linial semidefinite relaxation also has integrality gaps of order at least $`\\sqrt{\\log n}/(\\log\\log n)^3`$, approaching the square-root-logarithmic upper bound.", "coveredBy": [ "UniformSparsestCut" ], "note": "along a sequence n_j → ∞, OPT/GL >= c·sqrt(log n)/(log log n)^3 eventually; 'approaching the ... upper bound' cites ARV" } ], "gap": "constant-factor NP-hardness of Uniform Sparsest Cut", "scopeNoteOnGap": "The result is an existential sequence, not a bound for every size or a computational hardness theorem. (docs/117.md:9)", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "UniformSparsestCut", "flag": "JUNK? cleared: glValue > 0 is asserted for every j, so the ratio never divides by zero; log log n <= 0 for small n only weakens finitely many terms of an eventually-statement" } ], "flagReadings": [ { "challenge": "UniformSparsestCut", "flag": "JUNK? cleared: glValue > 0 is asserted for every j, so the ratio never divides by zero; log log n <= 0 for small n only weakens finitely many terms of an eventually-statement", "decision": "cleared", "why": "glValue is an sInf of values Σ cap·d ≥ 0, a set that is bounded below and nonempty for n ≥ 2 (e.g. a normalised cut metric). Its only junk value is 0, which the conjunct ∀ j, 0 < glValue (C j) excludes. So the ratio's denominator is the true positive relaxation optimum. OPT is a min over the finitely many nonempty proper B (n ≥ 2), whose cutRatio denominators |B|(n−|B|) > 0 are genuine. On the left side, (log log n)³ is 0 only if log n ∈ {0, ±1}, never for an integer n ≥ 2; it is negative only for n = 2, which makes that instance trivially true. Tendsto n atTop atTop puts n j ≥ 16 (log log > 0) for all large j, and the claim is ∀ᶠ j, so those cases are excluded from what is asserted. Normalised cut metrics are feasible, so glValue ≤ OPT and the gap is a genuine integrality gap. Lane K: CERTIFIED.", "evidence": [ "ComparatorChallenges/UniformSparsestCut.lean:20-25: noncomputable def cutRatio {n : ℕ} (C : Capacity n) (B : Finset (Fin n)) : ℝ := ⏎ (∑ i ∈ B, ∑ j ∈ Bᶜ, C.cap i j) / ⏎ ((B.card : ℝ) * ((n : ℝ) - (B.card : ℝ))) ⏎ ⏎ noncomputable def OPT {n : ℕ} (C : Capacity n) : ℝ := ⏎ sInf {r | ∃ B : Finset (Fin n), B.Nonempty ∧ B ≠ Finset.univ ∧ r = cutRatio C B}", "ComparatorChallenges/UniformSparsestCut.lean:27-36: def NegativeType {n : ℕ} (d : Fin n → Fin n → ℝ) : Prop := ⏎ (∃ x : Fin n → EuclideanSpace ℝ (Fin n), ∀ i j, d i j = ‖x i - x j‖ ^ 2) ∧ ⏎ ∀ i j k, d i k ≤ d i j + d j k ⏎ ⏎ def Feasible {n : ℕ} (d : Fin n → Fin n → ℝ) : Prop := ⏎ NegativeType d ∧ pairSum d = 1 ⏎ ⏎ noncomputable def glValue {n : ℕ} (C : Capacity n) : ℝ := ⏎ sInf {r | ∃ d : Fin n → Fin n → ℝ, Feasible d ∧ ⏎ r = pairSum (fun i j => C.cap i j * d i j)}", "ComparatorChallenges/UniformSparsestCut.lean:40-48: theorem mainGap : ⏎ ∃ c : ℝ, 0 < c ∧ ⏎ ∃ n : ℕ → ℕ, ∃ C : (j : ℕ) → Capacity (n j), ⏎ (∀ j, 2 ≤ n j) ∧ ⏎ Tendsto n atTop atTop ∧ ⏎ (∀ j, 0 < glValue (C j)) ∧ ⏎ ∀ᶠ j in atTop, ⏎ c * Real.sqrt (Real.log (n j : ℝ)) / (Real.log (Real.log (n j : ℝ))) ^ 3 ≤ ⏎ OPT (C j) / glValue (C j) := by" ] } ], "readings": [ "first/batch8.json", "flags/flags4.json" ] }, { "family": "118", "title": "Bin packing and unbounded configuration-LP gaps", "headline": "Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds 1/6 and each bin holds at most five items.", "challenges": [ "BinPackingGap" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant.", "coveredBy": [ "BinPackingGap" ], "note": "∀ c ∃ instance (5B items, sizes in (1/6,1)) with both configuration-LP values = B and opt > B + c." }, { "clause": "Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds 1/6 and each bin holds at most five items.", "coveredBy": [ "BinPackingGap" ], "note": "∀ c, every NP language Karp-reduces (by a polynomial-time TM2 machine) to instances with 5B items each > 1/6, mapping yes-instances to packable-in-B and no-instances to not-packable-in-B+c." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "BinPackingGap", "flag": "VACUITY checked, none: NPVerifier is inhabited (a constant verifier is poly-time computable), so InNP is not empty and PackingGapNPHard is not vacuously true" } ], "flagReadings": null, "readings": [ "first/batch6.json" ] }, { "family": "119", "title": "The Courtade–Kumar and Hellinger conjectures", "headline": "Proves the Courtade–Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise. A stronger theorem treats randomized binary summaries at fixed initial information. The Hellinger conjecture is also proved for every Boolean output bias and noise correlation.", "challenges": [ "CourtadeKumar", "SoftChannel204" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves the Courtade–Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise.", "status": "stated", "coveredBy": [ "CourtadeKumar", "SoftChannel204" ], "evidence": [ "lean/ComparatorChallenges/CourtadeKumar.lean:55: def CourtadeKumarStatement : Prop :=", "lean/ComparatorChallenges/CourtadeKumar.lean:56: ∀ (n : ℕ), 1 ≤ n →", "lean/ComparatorChallenges/CourtadeKumar.lean:57: ∀ ε : ℝ, 0 ≤ ε → ε ≤ (1 : ℝ) / 2 →", "lean/ComparatorChallenges/CourtadeKumar.lean:58: ∀ f : Cube n → Bool, mutualInformation ε f ≤ 1 - binaryEntropy ε", "lean/ComparatorChallenges/CourtadeKumar.lean:60: def DictatorAttainmentStatement : Prop :=", "lean/ComparatorChallenges/CourtadeKumar.lean:61: ∀ (n : ℕ), 1 ≤ n →", "lean/ComparatorChallenges/CourtadeKumar.lean:62: ∀ ε : ℝ, 0 ≤ ε → ε ≤ (1 : ℝ) / 2 →", "lean/ComparatorChallenges/CourtadeKumar.lean:63: ∀ i : Fin n,", "lean/ComparatorChallenges/CourtadeKumar.lean:64: mutualInformation ε (dictator i) = 1 - binaryEntropy ε ∧", "lean/ComparatorChallenges/CourtadeKumar.lean:65: mutualInformation ε (complementDictator i) = 1 - binaryEntropy ε", "lean/ComparatorChallenges/CourtadeKumar.lean:69: theorem courtadeKumarAndAttainment : FullStatement := by" ] }, { "clause": "A stronger theorem treats randomized binary summaries at fixed initial information.", "status": "stated", "coveredBy": [ "SoftChannel204" ], "evidence": [ "lean/ComparatorChallenges/SoftChannel204.lean:77: def SoftContraction : Prop := ∀ (n : ℕ) (u : Cube n → ℝ), Closed u →", "lean/ComparatorChallenges/SoftChannel204.lean:78: ∀ ρ ∈ Set.Icc (-1 : ℝ) 1,", "lean/ComparatorChallenges/SoftChannel204.lean:79: info (noise ρ u) ≤ psi (|ρ| * psiInv (info u))", "lean/ComparatorChallenges/SoftChannel204.lean:86: def SoftInformationIdentity : Prop := ∀ (n : ℕ) (u : Cube n → ℝ), Closed u →", "lean/ComparatorChallenges/SoftChannel204.lean:87: ∀ ρ ∈ Set.Icc (-1 : ℝ) 1,", "lean/ComparatorChallenges/SoftChannel204.lean:88: mutualInformation (softJoint ρ u) = info (noise ρ u)" ] }, { "clause": "The Hellinger conjecture is also proved for every Boolean output bias and noise correlation.", "status": "unlinked", "coveredBy": [], "manuscript": "Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026", "evidence": [ "corpus/openai-math/release.json:7916: \"dir\": \"Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026\"", "corpus/openai-math/release.json:7918: We prove the Hellinger conjecture for Boolean functions on the uniform discrete cube, with arbitrary output bias.", "lean/docs/119.md:3: The following describes the scope of the Lean formalization related to the following accompanying paper(s):", "lean/docs/119.md:5: - [Sharp binary-information contraction on the discrete cube](../../preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026/main.pdf)", "lean/docs/119.md:1: # The Courtade–Kumar and Hellinger conjectures", "lean/ComparatorChallenges/CourtadeKumar.lean, SoftChannel204.lean: no Hellinger functional (no sqrt(1-m^2) term) in either statement" ] } ], "gap": "The Hellinger conjecture (the October/September Hellinger paper's main theorem) is not stated anywhere in the family's challenges.", "scopeNoteOnGap": "silent. The note's title is \"The Courtade–Kumar and Hellinger conjectures\" (lean/docs/119.md:1), but it links only the binary-information paper (line 5) and never mentions Hellinger in its scope text (lines 9–11).", "note": "The two information-contraction clauses are stated (CourtadeKumar; SoftChannel204 SoftContraction/SoftAttainment/SoftInformationIdentity). The Hellinger clause is not stated and comes from the Hellinger manuscript, which the scope note does not link (only its title mentions Hellinger), hence UNLINKED; I differ from the first reader only through amendment 3.", "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "SoftChannel204", "flag": "JUNK? checked, cleared: F(s) = ψ⁻¹(s)·artanh(ψ⁻¹(s)) would give artanh(1) = 0 (log(2/0) = log 0 = 0) at s = log 2. SharpProduction assumes Interior g, which forces info g < log 2, so that endpoint is never reached. ψ⁻¹ is applied only to info values in [0, log 2] = ψ([0,1])." } ], "flagReadings": null, "readings": [ "first/batch5.json", "second/second1.json" ] }, { "family": "121", "title": "Almost-linear approximation of edit distance", "headline": "For every fixed rational $`\\varepsilon\\in(0,1)`$, gives a randomized $`(1+\\varepsilon)`$ approximation to unit-cost edit distance in worst-case expected time $`N^{1+o(1)}`$, with success probability at least 2/3. The strings have total length N and polynomially bounded integer symbols. This is an asymptotic guarantee at fixed accuracy.", "challenges": [ "EditApproximation" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "For every fixed rational $`\\varepsilon\\in(0,1)`$, gives a randomized $`(1+\\varepsilon)`$ approximation to unit-cost edit distance in worst-case expected time $`N^{1+o(1)}`$, with success probability at least 2/3.", "coveredBy": [ "EditApproximation" ], "note": "ε a BinaryFraction in (0,1) fixed first; for every input and every N >= |s|+|t|, P(edit <= answer <= (1+ε)·edit) >= 2/3 under the algorithm's own outcome law; for every C and η>0, eventually in N, worst-case (over inputs) expected work <= (N+2)^(1+η) — exactly N^{1+o(1)}" }, { "clause": "The strings have total length N and polynomially bounded integer symbols.", "coveredBy": [ "EditApproximation" ], "note": "List ℤ with |s|+|t| <= N; the work clause assumes |z| <= (N+2)^C for each fixed C; the correctness clause needs no symbol bound (stronger)" }, { "clause": "This is an asymptotic guarantee at fixed accuracy.", "coveredBy": [ "EditApproximation" ], "note": "∀ᶠ N with the threshold depending on ε, C, η" } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": "The statement's 'time' is a self-defined operation count of a statement-defined algorithm; the one point that is not visible in the statement is that the kernel work charged (from the rational twin program) equals the work of the bit program whose output is the answer — a second reader or lane K should confirm the solution-tree lemmas computedQueryMemoResult_decode / _localWork are sorry-free and used. Also: at every physically realizable N the formal algorithm is the exact quadratic DP (the fast branch needs log log N >= 2^1000), which is honest for an asymptotic claim but worth stating on the page.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [ { "challenge": "EditApproximation", "flag": "LARGE STATEMENT: 19,170 lines, ~1,156 declarations, essentially all in the theorem's closure; read by a delegated closure reader (whole closure, with mechanical scans of every *WithWork body) and spot-checked here" }, { "challenge": "EditApproximation", "flag": "DEFINITION CARRIES? (low, mitigated): the kernel work is charged from the exact-rational twin program (computedFamilyKernelWork, line 14788, sums rankedMemoKernelWork over physicalFamilyProgram) while the answer is produced by evaluating the rounded bit program (computedQueryMemoResultRawBand, line 17604, rankedBitMemoEvaluate). That the two visit the same fresh keys — i.e. that the charged work is the work of the computation that yields the answer — is not visible in the statement; it is stated by solution-tree lemmas (OAI/Combinatorics/EditApproximation/Refinement/ComputedQueryExecution.lean:37 computedQueryMemoResult_decode, :55 computedQueryMemoResult_localWork), read but not run" }, { "challenge": "EditApproximation", "flag": "JUNK? cleared: the expected work is a Bochner integral, but the outcome space is countable with all sets measurable and the work integrable (geometric failure counts with p >= 1/2 enter linearly), so it is the true expectation, not a junk 0; no Real.log, positive denominators, guarded ℕ subtractions" }, { "challenge": "EditApproximation", "flag": "QUANTIFIER? cleared: ε is fixed first, the work bound is ∀ C ∀ η ∃ threshold ∀ N >= threshold ∀ inputs (worst-case expected), matching N^{1+o(1)} at fixed accuracy" } ], "flagReadings": [ { "challenge": "EditApproximation", "flag": "DEFINITION CARRIES? (low, mitigated): the kernel work is charged from the exact-rational twin program (computedFamilyKernelWork, line 14788, sums rankedMemoKernelWork over physicalFamilyProgram) while the answer is produced by evaluating the rounded bit program (computedQueryMemoResultRawBand, line 17604, rankedBitMemoEvaluate). That the two visit the same fresh keys — i.e. that the charged work is the work of the computation that yields the answer — is not visible in the statement; it is stated by solution-tree lemmas (OAI/Combinatorics/EditApproximation/Refinement/ComputedQueryExecution.lean:37 computedQueryMemoResult_decode, :55 computedQueryMemoResult_localWork), read but not run", "decision": "cleared", "why": "The mechanism is as the first reader describes. The statement charges kernel work by walking the rational twin's memo traversal (rankedMemoKernelWork over physicalFamilyProgram, 14791-14793). Each fresh key costs the bit program's own per-key local work (cost = integerQueryLocalWorkRawBand = rankedBitLocalWork of the normalized bit program, 17616, passed in at 18174), while the answer comes from rankedBitMemoEvaluate (17607). The risk is that the two traversals visit different keys, so the charged work would not be the work of the computation that yields the answer. That risk does not materialize. In the solution tree, computedQueryMemoResult_decode proves that the executed bit memo result decodes to the rational result (ComputedQueryExecution.lean:37-39). decode keeps freshKeys, dictionaryVisits and requests unchanged (BitMemoRun.lean:17), so the bit program visits exactly the twin's fresh keys and dictionary entries. computedQueryMemoResult_localWork proves the bit program's own localWork equals computedFamilyKernelWork with that cost (55-57). RawBandPipelineBridges.lean:86 identifies the RawBand definitions in the statement with these. The hypotheses (2^1000 ≤ inputSmallLog N, the Nat.size mass bound, symbol and position bits, the occurrence table) are those of the large branch (physicalLargeInput, 15057-15061), the only branch where the twin is charged; the small branch charges chargedSuffixDP directly. So the statement's work is, as a theorem, the answer-producing program's own work. Caveats: I read the lemmas but did not run them. grep finds no sorry or axiom under OAI/Combinatorics/EditApproximation, and all three files lie in the import closure of RawBandMain.lean, so lane K's run of EditApproximation (not yet run) will kernel-check them.", "evidence": [ "lean/ComparatorChallenges/EditApproximation.lean:14791: OAI.EditApproximation.rankedMemoKernelWork (OAI.EditApproximation.SourceRanked.rank OAI.EditApproximation.ChargedPhysicalRequest.rank)", "lean/ComparatorChallenges/EditApproximation.lean:14793: (OAI.EditApproximation.physicalFamilyProgram source target N (OAI.EditApproximation.integerParameters N).P (OAI.EditApproximation.inputSmallLog N)", "lean/ComparatorChallenges/EditApproximation.lean:17607: OAI.EditApproximation.rankedBitMemoEvaluate (OAI.EditApproximation.SourceRanked.rank OAI.EditApproximation.ChargedPhysicalRequest.rank)", "lean/ComparatorChallenges/EditApproximation.lean:17616: OAI.EditApproximation.rankedBitLocalWork (OAI.EditApproximation.SourceRanked.rank OAI.EditApproximation.ChargedPhysicalRequest.rank)", "lean/ComparatorChallenges/EditApproximation.lean:18174: (OAI.EditApproximation.integerQueryLocalWorkRawBand source target N) (OAI.EditApproximation.integerPreevaluationWork source target N epsilon)", "lean/OAI/Combinatorics/EditApproximation/Refinement/ComputedQueryExecution.lean:37: theorem computedQueryMemoResult_decode :", "lean/OAI/Combinatorics/EditApproximation/Refinement/ComputedQueryExecution.lean:39: computedFamilyResult source target N input := by", "lean/OAI/Combinatorics/EditApproximation/Refinement/ComputedQueryExecution.lean:55: theorem computedQueryMemoResult_localWork :", "lean/OAI/Combinatorics/EditApproximation/Refinement/ComputedQueryExecution.lean:56: (computedQueryMemoResult source target N occurrence symbolBits positionBits input).localWork =", "lean/OAI/Combinatorics/EditApproximation/Refinement/ComputedQueryExecution.lean:57: computedFamilyKernelWork source target N input", "lean/OAI/Combinatorics/EditApproximation/Binary/BitMemoRun.lean:17: ⟨result.value.value, result.memory.mapValues BinaryFraction.value,", "lean/OAI/Combinatorics/EditApproximation/Specification/RawBandPipelineBridges.lean:86: @[simp] theorem OAI.EditApproximation.computedQueryMemoResultRawBand_eq : @OAI.EditApproximation.computedQueryMemoResultRawBand = @OAI.EditApproximation.computedQueryMemoResult := by" ] }, { "challenge": "EditApproximation", "flag": "JUNK? cleared: the expected work is a Bochner integral, but the outcome space is countable with all sets measurable and the work integrable (geometric failure counts with p >= 1/2 enter linearly), so it is the true expectation, not a junk 0; no Real.log, positive denominators, guarded ℕ subtractions", "decision": "cleared", "why": "This junk would point in the weakening direction: if the work were not integrable, the Bochner integral would be 0 and the work bound trivially true. The work is integrable. The outcome space is Unit ⊕ (coarse draws × (ι → ℕ) × accepted values). Here ι is Fintype, the coarse draws are finite (CoarseThresholdDraws is built from Fin B → Fin B), and the accepted values lie in Fin(range i). So the space is countable with measurable singletons, and the work is measurable. The only unbounded coordinates are the rejection failure counts. They are geometric with success parameter range/2^(clog₂ range) > 1/2 (rejectionSuccessParameter is N/R, with R < 2N proved at 14926-14941). They enter only through rejectionSelectedBits, linearly as (failures + 1)·bits; every other term is a function of the seed table and the accepted values, because physicalFamilyRuntimeAccepted keeps only execution.1 and execution.2.2. So the expected work is finite and the integral is the true expectation. Outside physicalLargeInput the law is a Dirac mass.", "evidence": [ "lean/ComparatorChallenges/EditApproximation.lean:1288: | D + 1 => (Fin B → Fin B) × (Fin B → CoarseThresholdDraws B D)", "lean/ComparatorChallenges/EditApproximation.lean:14943: geometricMeasure (OAI.EditApproximation.rejectionSuccessParameter (range i) (2 ^ OAI.EditApproximation.rejectionBitCount (range i))", "lean/ComparatorChallenges/EditApproximation.lean:14962: ∑ i, (((execution.1 i : ℝ) + 1) * OAI.EditApproximation.rejectionBitCount (range i)) *", "lean/ComparatorChallenges/EditApproximation.lean:15046: (execution.1, execution.2.2)", "lean/ComparatorChallenges/EditApproximation.lean:15185: if OAI.EditApproximation.physicalLargeInput N epsilon then (OAI.EditApproximation.computedExecutionLaw source target N).map Sum.inr", "lean/ComparatorChallenges/EditApproximation.lean:19159: (∫ outcome, OAI.EditApproximation.integerBinaryWorkRawBand source target N epsilon outcome" ] }, { "challenge": "EditApproximation", "flag": "QUANTIFIER? cleared: ε is fixed first, the work bound is ∀ C ∀ η ∃ threshold ∀ N >= threshold ∀ inputs (worst-case expected), matching N^{1+o(1)} at fixed accuracy", "decision": "cleared", "why": "ε is a theorem parameter, fixed first. The work conjunct reads ∀ C ∀ η > 0, ∀ᶠ N, ∀ inputs with |s|+|t| ≤ N and symbols ≤ (N+2)^C: E[work] ≤ (N+2)^(1+η). That is worst-case expected time N^(1+o(1)) at fixed accuracy for polynomially bounded symbols: the threshold may depend on (ε, C, η) but not on the input. The correctness conjunct holds for every N ≥ |s|+|t|, which includes N = |s|+|t|.", "evidence": [ "lean/ComparatorChallenges/EditApproximation.lean:19146: theorem almostLinearEditDistanceRawBand (epsilon : OAI.EditApproximation.BinaryFraction)", "lean/ComparatorChallenges/EditApproximation.lean:19156: (∀ (C : ℕ) (eta : ℝ), 0 < eta → ∀ᶠ N : ℕ in atTop,", "lean/ComparatorChallenges/EditApproximation.lean:19157: ∀ source target : List ℤ, source.length + target.length ≤ N →", "lean/ComparatorChallenges/EditApproximation.lean:19158: (∀ z ∈ source ++ target, z.natAbs ≤ (N + 2) ^ C) →", "lean/ComparatorChallenges/EditApproximation.lean:19161: (N + 2 : ℝ) ^ (1 + eta)) ∧" ] } ], "readings": [ "first/batch8.json", "flags/flags1.json" ] }, { "family": "122", "title": "Quantitative trace-reconstruction bounds with a uniform decoder", "headline": "At every fixed deletion probability in $`(0,1)`$, reconstructing an arbitrary length-n binary string requires $`n^{\\Omega(\\log\\log n)}`$ independent traces, ruling out polynomial-sample reconstruction. A uniform decoder achieves quasipolynomial sample and running-time bounds for known fixed rational retention probabilities. When the deletion probability is at most $`n^{-\\varepsilon}`$ for fixed ε > 0, both bounds become polynomial in the input and parameter encoding.", "challenges": [ "TraceReconstruction" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "At every fixed deletion probability in $`(0,1)`$, reconstructing an arbitrary length-n binary string requires $`n^{\\Omega(\\log\\log n)}`$ independent traces, ruling out polynomial-sample reconstruction.", "status": "stated", "coveredBy": [ "TraceReconstruction" ], "evidence": [ "lean/ComparatorChallenges/TraceReconstruction.lean:68: (hscale : Tendsto (fun k => (q k) ^ 3 * Real.log (n k)) atTop atTop) :", "lean/ComparatorChallenges/TraceReconstruction.lean:70: ENNReal.ofReal ((n k : ℝ) ^ (c * Real.log ((q k) ^ 3 * Real.log (n k)))) ≤", "lean/ComparatorChallenges/TraceReconstruction.lean:83: (fun n : ℕ => sampleComplexity n q s / ENNReal.ofReal ((n : ℝ) ^ C))", "lean/ComparatorChallenges/TraceReconstruction.lean:84: atTop (𝓝 ⊤) := by" ], "note": "With q_k = q fixed and n_k = k, the quantitative theorem gives sampleComplexity ≥ n^{c(log log n + 3 log q)} = n^{Ω(log log n)} by elementary asymptotics; superpolynomial_sample_complexity rules out polynomial samples. The estimator is an arbitrary randomized map on the complete traces, and success is worst-case over inputs at every success level s ∈ (0,1]. sInf ∅ = ⊤ only strengthens a lower bound, and for s < 1 the set of sufficient budgets is nonempty." }, { "clause": "A uniform decoder achieves quasipolynomial sample and running-time bounds for known fixed rational retention probabilities.", "status": "unlinked", "coveredBy": [], "manuscript": "Uniform-quasipolynomial-time-trace-reconstruction-October-5-2026", "evidence": [ "lean/docs/122.md:5: - [Quantitative lower bounds for trace reconstruction](../../preprints/quantitative-lower-bounds-for-trace-reconstruction-September-24-2026/paper.pdf)", "lean/docs/122.md:9: The formalization proves superpolynomial sample lower bounds for exact worst-case reconstruction of binary words from independent deletion traces", "CONTENTS.md:2988: We give a uniform algorithm that reconstructs every binary string from independent deletion traces when its length and rational retention probability are known. For each fixed retention probability, both the number of traces and the bit complexity are quasipolynomial in the string length." ], "note": "The challenge contains only lower bounds: no upper bound on sampleComplexity, no decoder, no running time. The sample-bound half also appears in the unlinked A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026." }, { "clause": "When the deletion probability is at most $`n^{-\\varepsilon}`$ for fixed ε > 0, both bounds become polynomial in the input and parameter encoding.", "status": "unlinked", "coveredBy": [], "manuscript": "Uniform-quasipolynomial-time-trace-reconstruction-October-5-2026", "evidence": [ "lean/docs/122.md:5: - [Quantitative lower bounds for trace reconstruction]", "CONTENTS.md:2988: If the deletion probability is at most $`n^{-\\varepsilon}`$ for fixed ε > 0, the sample and running-time bounds are polynomial." ], "note": "An upper-bound clause from the same unlinked manuscript." } ], "gap": "The headline's two upper-bound sentences (a uniform decoder with quasipolynomial sample and running-time bounds for fixed rational retention probability; polynomial bounds when the deletion probability is ≤ n^{-ε}) — the content of two of the family's three manuscripts — are not stated by any challenge. Only the lower bounds are.", "scopeNoteOnGap": "silent — docs/122.md lists only the lower-bound manuscript ('- [Quantitative lower bounds for trace reconstruction](…)', line 5) and describes only lower bounds (line 9: 'The formalization proves superpolynomial sample lower bounds for exact worst-case reconstruction of binary words from independent deletion traces…'); it never says the decoder / upper-bound results are left out.", "note": "The lower-bound sentence is stated. Both upper-bound sentences come from the two October-5 manuscripts, which docs/122.md does not link. The first reader asked whether listing only the lower-bound paper counts as declaring the decoder out; under amendment 3 it is UNLINKED (not DECLARED, since the note never names the omission).", "firstSuspicious": "The decision between DECLARED and UNDECLARED turns on whether listing only the lower-bound paper in docs/122.md counts as declaring the decoder out; I read 'says what is left out' strictly (it does not). Second reader please confirm.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json", "second/second2.json" ] }, { "family": "124", "title": "Polynomial-time scheduling on three identical machines", "headline": "Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines. For an explicitly given precedence graph, it decides deadline feasibility exactly and constructs a feasible schedule.", "challenges": [ "ThreeMachine" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines.", "coveredBy": [ "ThreeMachine" ], "note": "Deadline-free input: the machine outputs a feasible schedule whose makespan is minimal." }, { "clause": "For an explicitly given precedence graph, it decides deadline feasibility exactly and constructs a feasible schedule.", "coveredBy": [ "ThreeMachine" ], "note": "Deadline input T: outputs 'none' iff no feasible schedule, else a feasible schedule." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch1.json" ] }, { "family": "125", "title": "The metric k-median approximation threshold and recovery", "headline": "Gives a deterministic polynomial-time $`(1+2/e+\\varepsilon)`$-approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε > 0. Assuming $`P\\ne NP`$, the optimal infimum approximation factor is $`1+2/e`$.", "challenges": [ "KMedianRecovery", "KMedianRefinedRecovery", "KMedianThreshold", "MetricKMedian" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Gives a deterministic polynomial-time $`(1+2/e+\\varepsilon)`$-approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε > 0.", "coveredBy": [ "MetricKMedian" ], "note": "∀ ε>0 ∃ algorithm with a polynomial-time TM2 machine (all stack alphabets finite) outputting the membership mask of a nonempty facility set of size <= k with cost <= (1+2/e+ε)·OPT, for every instance" }, { "clause": "Assuming $`P\\ne NP`$, the optimal infimum approximation factor is $`1+2/e`$.", "coveredBy": [ "KMedianThreshold" ], "note": "under the challenge's own P ≠ NP (finite-alphabet TM2 polytime deciders vs verifiers), sInf of achievable deterministic polytime factors = 1+2/e" } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "MetricKMedian", "flag": "JUNK? cleared: optimum's else-branch (0) is unreachable because k>=1 and |facilities|>=k make feasibleSets nonempty" }, { "challenge": "KMedianThreshold", "flag": "VACUITY? (low, standard): the theorem is conditional on the file's bespoke PNeNP; if that bespoke separation were provably false the statement would be vacuous — the definitions read as the standard ones" }, { "challenge": "KMedianThreshold", "flag": "JUNK? cleared: approximationFactors is nonempty (by MetricKMedian) and bounded below by 1 (instances with positive optimum), so sInf is meaningful" }, { "challenge": "KMedianRefinedRecovery", "flag": "META: the challenge file runs a metaprogram that resets the auxiliary-lemma environment extension: line 61 `run_elab Lean.modifyEnv fun environment => Lean.Meta.auxLemmasExt.setState environment {}` — not mathematical, but an environment-modifying command inside a statement file deserves lane K's attention (same pattern appears in at least nine other challenge files)" } ], "flagReadings": [ { "challenge": "MetricKMedian", "flag": "JUNK? cleared: optimum's else-branch (0) is unreachable because k>=1 and |facilities|>=k make feasibleSets nonempty", "decision": "cleared", "why": "k_pos (1 ≤ k) and k_le (k ≤ facilities.card) make facilities nonempty. Any singleton {f} ⊆ facilities is then nonempty with card 1 ≤ k, so it lies in feasibleSets. feasibleSets is therefore nonempty and optimum is the genuine minimum cost. nearest's else-branch is unreachable for the sets Correct quantifies over, because Correct requires S.Nonempty. If optimum had taken the junk value 0, Correct would demand cost ≤ 0, which is harder, not weaker.", "evidence": [ "lean/ComparatorChallenges/MetricKMedian.lean:22: k_pos : 1 ≤ k", "lean/ComparatorChallenges/MetricKMedian.lean:23: k_le : k ≤ facilities.card", "lean/ComparatorChallenges/MetricKMedian.lean:39: if h : S.Nonempty then", "lean/ComparatorChallenges/MetricKMedian.lean:47: I.facilities.powerset.filter (fun S => S.Nonempty ∧ S.card ≤ I.k)", "lean/ComparatorChallenges/MetricKMedian.lean:50: if h : (feasibleSets I).Nonempty then", "lean/ComparatorChallenges/MetricKMedian.lean:51: ((feasibleSets I).image (cost I)).min' (h.image _)", "lean/ComparatorChallenges/MetricKMedian.lean:52: else 0", "lean/ComparatorChallenges/MetricKMedian.lean:55: ∃ S : Finset (Fin I.n), S ⊆ I.facilities ∧ S.Nonempty ∧ S.card ≤ I.k ∧", "lean/ComparatorChallenges/MetricKMedian.lean:57: cost I S ≤ (1 + 2 / Real.exp 1 + epsilon) * optimum I" ] }, { "challenge": "KMedianThreshold", "flag": "VACUITY? (low, standard): the theorem is conditional on the file's bespoke PNeNP; if that bespoke separation were provably false the statement would be vacuous — the definitions read as the standard ones", "decision": "cleared", "why": "The bespoke classes read as the standard ones. FinitePolyTime is Mathlib's TM2ComputableInPolyTime (a polynomial step bound in input length) plus finiteness of every stack alphabet. Mathlib's FinTM2 already forces finite K, Λ and σ but only a finite input alphabet, so the added condition is what makes it a genuine finite multistack machine. P is the languages decided in polynomial time on the raw word. NP is a polynomial-length witness checked by a poly-time verifier on the self-delimiting encodePair. With these definitions P = NP is exactly the open problem and nothing makes it provable, so the hypothesis is not refutable. The conditional form matches the prose ('Assuming P ≠ NP', docs/125.md:16).", "evidence": [ "lean/ComparatorChallenges/KMedianThreshold.lean:65: ∃ p : Turing.TM2ComputableInPolyTime ea eb f,", "lean/ComparatorChallenges/KMedianThreshold.lean:66: ∀ k : p.tm.K, Finite (p.tm.Γ k)", "lean/ComparatorChallenges/KMedianThreshold.lean:76: def P : Set Language := {L | ∃ decideL : Word → Bool,", "lean/ComparatorChallenges/KMedianThreshold.lean:77: FinitePolyTime id encodeBit decideL ∧ ∀ x, decideL x = true ↔ x ∈ L}", "lean/ComparatorChallenges/KMedianThreshold.lean:79: def NP : Set Language := {L | ∃ (verify : Word × Word → Bool) (p : Polynomial ℕ),", "lean/ComparatorChallenges/KMedianThreshold.lean:80: FinitePolyTime encodePair encodeBit verify ∧", "lean/ComparatorChallenges/KMedianThreshold.lean:81: ∀ x, x ∈ L ↔ ∃ w : Word, w.length ≤ p.eval x.length ∧ verify (x, w) = true}", "lean/ComparatorChallenges/KMedianThreshold.lean:84: def PNeNP : Prop := P ≠ NP", "lean/ComparatorChallenges/KMedianThreshold.lean:102: theorem threshold (h : Complexity.PNeNP) :", "lean/docs/125.md:16: Assuming $P\\ne NP$, it also proves that the infimum of all polynomial-time approximation factors in this model is exactly $1+2/e$." ] }, { "challenge": "KMedianThreshold", "flag": "JUNK? cleared: approximationFactors is nonempty (by MetricKMedian) and bounded below by 1 (instances with positive optimum), so sInf is meaningful", "decision": "cleared", "why": "approximationFactors ⊆ [1, ∞). Take the instance n = 2, d(0,1) = 1, clients {0}, facilities {1}, k = 1. Its optimum is 1, and its only allowed output S = {1} costs 1, so FactorCorrect α forces α ≥ 1; a negative α is impossible too. The set is nonempty: deterministic poly-time constant-factor algorithms exist mathematically, and MetricKMedian.main_finite_work supplies 1 + 2/e + ε. Nonempty and bounded below, so sInf is the genuine infimum; the set is upward-closed because optimum ≥ 0. If the set were empty, sInf = 0 ≠ 1 + 2/e, so a junk value could only falsify the statement.", "evidence": [ "lean/ComparatorChallenges/KMedianThreshold.lean:91: cost I S ≤ α * optimum I", "lean/ComparatorChallenges/KMedianThreshold.lean:94: def approximationFactors : Set ℝ := {α | ∃ algorithm : Instance → List Bool,", "lean/ComparatorChallenges/KMedianThreshold.lean:95: Complexity.FinitePolyTime encodeInstance id algorithm ∧", "lean/ComparatorChallenges/KMedianThreshold.lean:96: ∀ I, FactorCorrect α I (algorithm I)}", "lean/ComparatorChallenges/KMedianThreshold.lean:103: sInf approximationFactors=1+2/Real.exp 1 := by" ] } ], "readings": [ "first/batch8.json", "flags/flags2.json" ] }, { "family": "126", "title": "Exponential semidefinite complexity of perfect matching", "headline": "Proves that every exact semidefinite lift of the perfect matching polytope has exponential size, answering Rothvoss's polynomial-size lift question negatively. The bound holds even for the positive semidefinite rank of its odd-cut slack matrix after any fixed shift $`0\\lt \\rho\\lt 1`$, allowing arbitrary real positive semidefinite factors.", "challenges": [ "MatchingAffineLift", "MatchingPSD" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that every exact semidefinite lift of the perfect matching polytope has exponential size, answering Rothvoss's polynomial-size lift question negatively.", "coveredBy": [ "MatchingAffineLift" ], "note": "Partial: the Lean states SUPERPOLYNOMIAL size (∀ C > 0, eventually for even n, every affine spectrahedral lift has matrix size r > n^C), not exponential. Superpolynomial already answers Rothvoss's polynomial-size question negatively." }, { "clause": "The bound holds even for the positive semidefinite rank of its odd-cut slack matrix after any fixed shift $`0\\lt \\rho\\lt 1`$, allowing arbitrary real positive semidefinite factors.", "coveredBy": [], "note": "Not stated: MatchingPSD bounds the PSD rank of the UNSHIFTED slack matrix (entries |M ∩ δ(U)| − 1, plus edge rows), superpolynomially; no shift ρ appears. Arbitrary real PSD factors are allowed (no equivariance or rank-one restriction)." } ], "gap": "Exponential (2^{Ω(n)}) lower bounds and the shifted matrices (0 < ρ < 1) are not stated; only superpolynomial bounds for the unshifted slack matrix and for affine lifts.", "scopeNoteOnGap": "lean/docs/126.md:11: \"These selected statements give superpolynomial growth. They do not state the paper's exponential bound for every fixed positive shift.\"", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "MatchingPSD", "flag": "JUNK? checked, none: if the set were empty, psdRank = 0 would make the claim false, not trivial." } ], "flagReadings": [ { "challenge": "MatchingPSD", "flag": "JUNK? checked, none: if the set were empty, psdRank = 0 would make the claim false, not trivial.", "decision": "cleared", "why": "Nat sInf ∅ = 0 would turn the claim into n^C < 0, which is false, so the junk direction is harmless. It is not reached either: for even n every slack entry is ≥ 0 (edge rows are 0/1, and a perfect matching crosses every odd set U at least once, because matching edges inside U cover an even number of U's vertices), so with r = number of perfect matchings (> 0 for even n) the diagonal factorization A_i = diag(slack i M)_M, B_M = E_MM gives trace(A_i B_M) = slack i M. The set is nonempty and psdRank is the genuine least positive order. Lane K: CERTIFIED.", "evidence": [ "ComparatorChallenges/MatchingPSD.lean:27: noncomputable def slack {n : ℕ} (i : Row n) (M : PerfectMatching n) : ℝ := by", "ComparatorChallenges/MatchingPSD.lean:28: classical", "ComparatorChallenges/MatchingPSD.lean:29: exact match i with", "ComparatorChallenges/MatchingPSD.lean:30: | Sum.inl e => if e ∈ M.1 then 1 else 0", "ComparatorChallenges/MatchingPSD.lean:31: | Sum.inr U => (crossingCount U.1 M : ℝ) - 1", "ComparatorChallenges/MatchingPSD.lean:34: def HasFactorization (n r : ℕ) : Prop :=", "ComparatorChallenges/MatchingPSD.lean:35: ∃ (A : Row n → Matrix (Fin r) (Fin r) ℝ)", "ComparatorChallenges/MatchingPSD.lean:36: (B : PerfectMatching n → Matrix (Fin r) (Fin r) ℝ),", "ComparatorChallenges/MatchingPSD.lean:37: (∀ i, (A i).PosSemidef) ∧ (∀ M, (B M).PosSemidef) ∧", "ComparatorChallenges/MatchingPSD.lean:38: ∀ i M, slack i M = Matrix.trace (A i * B M)", "ComparatorChallenges/MatchingPSD.lean:41: noncomputable def psdRank (n : ℕ) : ℕ :=", "ComparatorChallenges/MatchingPSD.lean:42: sInf {r : ℕ | 0 < r ∧ HasFactorization n r}", "ComparatorChallenges/MatchingPSD.lean:46: def MainClaim : Prop :=", "ComparatorChallenges/MatchingPSD.lean:47: ∀ C : ℝ, 0 < C → ∃ n₀ : ℕ, 4 ≤ n₀ ∧", "ComparatorChallenges/MatchingPSD.lean:48: ∀ n : ℕ, n₀ ≤ n → Even n → (n : ℝ) ^ C < (psdRank n : ℝ)" ] } ], "readings": [ "first/batch5.json", "flags/flags3.json" ] }, { "family": "127", "title": "Average sensitivity of polynomial threshold functions", "headline": "Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most $`8d\\sqrt n`$, uniformly for $`1\\le d\\le n`$. Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with $`\\mathop{\\mathrm{sign}}\\nolimits (0)=1`$.", "challenges": [ "GotsmanLinial" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most $`8d\\sqrt n`$, uniformly for $`1\\le d\\le n`$.", "coveredBy": [ "GotsmanLinial" ], "note": "∀ n, d with 1 ≤ d ≤ n, ∀ real multilinear p of total degree ≤ d: AS(sign∘p) ≤ 8d√n." }, { "clause": "Average sensitivity counts expected output changes under single-bit flips.", "coveredBy": [ "GotsmanLinial" ], "note": "averageSensitivity = Σ_i #{x : f(x) ≠ f(x^{⊕i})}/2^n." }, { "clause": "This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with $`\\mathop{\\mathrm{sign}}\\nolimits (0)=1`$.", "coveredBy": [ "GotsmanLinial" ], "note": "thresholdSign t = 1 if 0 ≤ t else −1." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch5.json" ] }, { "family": "128", "title": "A factor-two approximation for shortest common superstring", "headline": "Gives a deterministic polynomial-time algorithm constructing a common superstring of length at most twice the optimum for every finite family of explicitly represented strings. The running time is polynomial in the full encoded input length, including symbol labels.", "challenges": [ "Superstring" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Gives a deterministic polynomial-time algorithm constructing a common superstring of length at most twice the optimum for every finite family of explicitly represented strings.", "coveredBy": [ "Superstring" ], "note": "∃ f with a Mathlib TM2 poly-time implementation such that f S is a common superstring (contiguous) and |f S| ≤ 2·opt S for every instance." }, { "clause": "The running time is polynomial in the full encoded input length, including symbol labels.", "coveredBy": [ "Superstring" ], "note": "Symbols are bit-lists, so labels are part of the encoded input; time is polynomial in the encoded length." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch5.json" ] }, { "family": "129", "title": "Exponential state costs for two-way automata", "headline": "Proves exponential lower bounds both for complementing two-way nondeterministic finite automata and for simulating one-way nondeterministic automata by two-way deterministic ones. The latter resolves the Sakoda–Sipser state-succinctness conjecture over growing finite alphabets; both results rule out polynomial state bounds independent of alphabet size.", "challenges": [ "TwoWayComplementation", "TwoWayDeterminization", "OneWayLiveness" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves exponential lower bounds both for complementing two-way nondeterministic finite automata", "coveredBy": [ "TwoWayComplementation", "TwoWayDeterminization" ], "note": "For every n >= 4 there is an n-state 2NFA A whose complement needs at least (1/2) 2^floor((n-4)/127) - 1 states, for every finite-state 2NFA." }, { "clause": "and for simulating one-way nondeterministic automata by two-way deterministic ones", "coveredBy": [ "OneWayLiveness" ], "note": "An (h+3)-state NFA with no left moves recognizes one-way liveness on h points, and every s-state 2DFA recognizing it has 2^floor((h-2)/31) <= 4(s+2)^2, or 4(s+1)^2 under the non-positive convention." }, { "clause": "The latter resolves the Sakoda–Sipser state-succinctness conjecture over growing finite alphabets", "coveredBy": [ "OneWayLiveness" ], "note": "Alphabet BRel (Fin h), finite and growing with h." }, { "clause": "both results rule out polynomial state bounds independent of alphabet size", "coveredBy": [ "TwoWayComplementation", "TwoWayDeterminization", "OneWayLiveness" ], "note": "Both bounds are exponential in the parameter that drives the alphabet size." } ], "gap": null, "scopeNoteOnGap": "no gap", "note": null, "firstSuspicious": "docs/129.md:12 says \"The separate results about one-way liveness and the Sakoda–Sipser problem are outside this scope\", but docs/129.md:14 and the OneWayLiveness challenge then cover exactly that result. The sentence refers to the first paper's scope. It is a wording inconsistency, not a gap.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "130", "title": "Exact Fourier transforms below $`n\\log n`$", "headline": "Gives a deterministic length-n discrete Fourier transform algorithm using $`O(n(\\log n)^{1-\\delta})`$ operations for every n, with explicit $`\\delta=10^{-13}`$. The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing.", "challenges": [ "ExactFourier", "UniformFourier" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Gives a deterministic length-n discrete Fourier transform algorithm using $`O(n(\\log n)^{1-\\delta})`$ operations for every n, with explicit $`\\delta=10^{-13}`$", "coveredBy": [ "UniformFourier" ], "note": "transform_main: one fixed program pair (order, solve) computes dft n x for every n > 0 and every x, with total work <= W n, where W = O(n (log n)^(1 - 10^-13)), W = O(paperTime) and W = o(n log n)." }, { "clause": "The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing", "coveredBy": [ "UniformFourier" ], "note": "Exact C arithmetic; the root zeta_d is supplied as an input, with d = run order n and 0 < d < 1024 n^3; preparation (order, scalars) is charged; every integer is bounded by (n+2)^cBound (logarithmic words). The model has no instruction for arbitrary complex constants (scalars are built from 0, 1 and zeta_d by +, -, *, inverse), which is more restrictive than \"unrestricted coefficients\", so the Lean claim is if anything stronger here." } ], "gap": null, "scopeNoteOnGap": "no gap", "note": null, "firstSuspicious": "UniformFourier's cost model is in-file and intricate (lines 118-254). It is worth one independent pass looking for an uncharged path, e.g. work hidden inside a Handler call or a Tape built through Atom.cz or Ty.blank at a nontrivial length.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "UniformFourier", "flag": "DEFINITION (not a hole): the operation-count semantics (Bill, Code.run, Atom.run) is defined in the challenge. An under-charged path would weaken the statement. My reading found none, but a second reader should audit Code.run for any free work." } ], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "131", "title": "Rapid mixing of graph switches for every degree sequence", "headline": "Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in $`O(n^8)`$ time for every graphical labeled degree sequence. The same degree-constrained graphs can also be sampled exactly uniformly by an almost-surely terminating algorithm with expected polynomial bit running time.", "challenges": [ "SwitchChain" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in $`O(n^8)`$ time for every graphical labeled degree sequence.", "coveredBy": [ "SwitchChain" ], "note": "For n ≥ 4: t_mix(1/4) ≤ 2n^8 for the lazy switch chain (1/2 hold + a uniform proposal over the 6·C(n,4) ordered pairs of distinct perfect matchings on 4-sets); plus connectivity, the spectral gap ≥ 1/(24n²C(n,4)) and an exponential TV bound. n ≤ 3 is excluded; every degree sequence there has a unique realization." }, { "clause": "The same degree-constrained graphs can also be sampled exactly uniformly by an almost-surely terminating algorithm with expected polynomial bit running time.", "coveredBy": [], "note": "Not stated; declared out of scope." } ], "gap": "The exactly uniform sampler is not stated (also n ≥ 4, a harmless restriction).", "scopeNoteOnGap": "lean/docs/131.md:11: \"The paper's exactly uniform sampling algorithm is outside these selected chain estimates.\"", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "SwitchChain", "flag": "JUNK? checked, none for n ≥ 4: the kernel divides by 12·C(n,4) > 0; mixingTime uses Nat.find under an existence proof that the statement itself supplies." } ], "flagReadings": [ { "challenge": "SwitchChain", "flag": "JUNK? checked, none for n ≥ 4: the kernel divides by 12·C(n,4) > 0; mixingTime uses Nat.find under an existence proof that the statement itself supplies.", "decision": "cleared", "why": "For n ≥ 4, 12·C(n,4) ≥ 12 > 0, so switchKernel has no division junk, and it is a genuine lazy kernel. There are 6·C(n,4) proposals (3 perfect matchings per 4-set, ordered distinct pairs), and each proposal p sends G to exactly one state (a valid switch preserves degrees; an invalid one stays at G), so each row sums to 1/2 + 6C(n,4)/(12C(n,4)) = 1. uniformAverage and tvDistanceFrom divide by stateCount ≥ 1 (Graphical = Nonempty). mixingTime is Nat.find under a proof of ∃ t, MixedAt that switch_chain_main itself asserts (∃ h, mixingTime n d h ≤ 2n^8). Nat.find is the least such t whichever proof is passed, so it is the true mixing time at distance 1/4. The '∀ h, mixingTime … = 0' clause is likewise proof-independent.", "evidence": [ "ComparatorChallenges/SwitchChain.lean:34-36: abbrev SwitchProposal (n : Nat) := ⏎ Σ S : FourSet n, ⏎ {p : PerfectMatching S × PerfectMatching S // p.1 ≠ p.2}", "ComparatorChallenges/SwitchChain.lean:87-94: def switchKernel (n : Nat) (d : Fin n → Nat) ⏎ (G H : GraphState n d) : ℝ := by ⏎ classical ⏎ exact ⏎ (if G = H then (1 : ℝ) / 2 else 0) + ⏎ (((allProposals n).filter ⏎ (fun p => proposalResult G.val p = H.val)).card : ℝ) / ⏎ (12 * (Nat.choose n 4 : ℝ))", "ComparatorChallenges/SwitchChain.lean:108-111: def uniformAverage (n : Nat) (d : Fin n → Nat) ⏎ (f : GraphState n d → ℝ) : ℝ := ⏎ Finset.sum (allGraphStates n d) (fun G => f G) / ⏎ (stateCount n d : ℝ)", "ComparatorChallenges/SwitchChain.lean:145-148: def mixingTime (n : Nat) (d : Fin n → Nat) ⏎ (h : ∃ t : Nat, MixedAt n d t) : Nat := by ⏎ classical ⏎ exact Nat.find h", "ComparatorChallenges/SwitchChain.lean:156: ∀ (n : Nat) (d : Fin n → Nat), 4 ≤ n → (∀ v, d v ≤ n - 1) → Graphical n d → ((∃ h : (∃ t : Nat, MixedAt n d t), mixingTime n d h ≤ 2 * n ^ 8) ∧ (stateCount n d = 1 → (MixedAt n d 0 ∧ ∀ h : (∃ t : Nat, MixedAt n d t), mixingTime n d h = 0)) ∧ (1 < stateCount n d → hasSpectralGapAtLeast n d ((1 : ℝ) / (24 * (n : ℝ) ^ 2 * (Nat.choose n 4 : ℝ))))) := by" ] } ], "readings": [ "first/batch5.json", "flags/flags4.json" ] }, { "family": "132", "title": "A superquadratic separation of sensitivity and block sensitivity", "headline": "Constructs total Boolean functions with block sensitivity $`\\mathop{\\mathrm{bs}}\\nolimits (f)\\ge s(f)^\\alpha`$ for a fixed α > 2, disproving the quadratic strengthening of the Sensitivity Conjecture. Here $`s(f)`$ counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together.", "challenges": [ "SensitivitySeparation" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Constructs total Boolean functions with block sensitivity $`\\mathop{\\mathrm{bs}}\\nolimits (f)\\ge s(f)^\\alpha`$ for a fixed α > 2", "status": "undeclared", "coveredBy": [], "manuscript": "A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026", "evidence": [ "lean/ComparatorChallenges/SensitivitySeparation.lean:34: (2 : ℝ) ^ d / (4 * ((d : ℝ) + 2) ^ 2) ≤", "lean/ComparatorChallenges/SensitivitySeparation.lean:35: (blockSensitivity f : ℝ) / (sensitivity f : ℝ) ^ 2 := by", "lean/ComparatorChallenges/SensitivitySeparation.json:5: \"OAI.Paper320.quantitative_separation\"", "lean/OAI/Combinatorics/Sensitivity/Separation.lean:36: theorem fixed_power_separation :", "lean/docs/132.md:9: The formalization also gives a fixed exponent $\\alpha>2$ and a sequence with $s(f)^\\alpha\\le\\mathrm{bs}(f,0)$ while the latter tends to infinity" ] }, { "clause": "disproving the quadratic strengthening of the Sensitivity Conjecture.", "status": "stated", "coveredBy": [ "SensitivitySeparation" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/SensitivitySeparation.lean:33: (∃ x y : Fin n → Bool, f x ≠ f y) ∧" ] }, { "clause": "Here $`s(f)`$ counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together.", "status": "stated", "coveredBy": [ "SensitivitySeparation" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/SensitivitySeparation.lean:15: (Finset.univ.filter fun i => f (flip x {i}) ≠ f x).card", "lean/ComparatorChallenges/SensitivitySeparation.lean:23: if (∀ A ∈ blocks, ∀ B ∈ blocks, A ≠ B → Disjoint A B) ∧" ] } ], "gap": "The fixed-exponent separation bs(f) ≥ s(f)^α with α > 2 (and unbounded bs) is claimed by the headline but not stated by any Comparator challenge. The scope note does not declare this omission; it asserts the opposite ('The formalization also gives a fixed exponent α>2 …', lean/docs/132.md:9), which is true of the formal library but not of the Comparator statement.", "scopeNoteOnGap": "The formalization also gives a fixed exponent $\\alpha>2$ and a sequence with $s(f)^\\alpha\\le\\mathrm{bs}(f,0)$ while the latter tends to infinity; here $\\mathrm{bs}(f,0)$ is block sensitivity at the all-zero input.", "note": "The one theorem gives bs(f)/s(f)² ≥ 2^d/(4(d+2)²) for each d (ratio unbounded, refuting bs ≤ C·s²; s ≥ 1 because f is nonconstant, so no x/0 junk), but nothing bounds s(f) in terms of d, so no fixed α > 2 follows. Flag CREDIT OUTSIDE THE CLOSURE: the note credits 'the formalization' with the α > 2 sequence, which is OAI.Paper320.fixed_power_separation in the solution library, not in the challenge's theorem_names, so Comparator never checks it. A credit is the opposite of naming an omission, so the clause is undeclared. Agrees with the first reader.", "firstSuspicious": "The gap is in what Comparator certifies, not necessarily in what was formalized: the α > 2 family appears as a separate theorem in the formal library (SeedAmplification.lean:14). The scope note presents it as part of 'the formalization' without saying it is outside the Comparator statement. A second reader should confirm the classification (UNDECLARED because the note claims the result rather than excluding it).", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "SensitivitySeparation", "flag": "NARROWER — UNDECLARED: the headline's 'bs(f) ≥ s(f)^α for a fixed α > 2' is absent from the Comparator statement, while lean/docs/132.md:9 says 'The formalization also gives a fixed exponent α>2 …'. That sentence describes the formal library (OAI/Combinatorics/Sensitivity/SeedAmplification.lean:14, theorem constructed_power_family), which lies outside the Comparator closure, so Comparator does not check it." }, { "challenge": "SensitivitySeparation", "flag": "JUNK? (checked, benign): division by s(f)² would be x/0 = 0 for constant f, but f is required nonconstant, so s(f) ≥ 1." } ], "flagReadings": null, "readings": [ "first/batch3.json", "second/second3.json" ] }, { "family": "133", "title": "The computational complexity of Weisfeiler–Leman refinement", "headline": "Proves unconditional $`n^{\\Omega(k)}`$ deterministic time lower bounds for joint and separate k-dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed k in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; deciding whether refinement identifies a graph is also EXPTIME-complete.", "challenges": [ "ParityLifts", "WLIdentification", "WeisfeilerLeman", "VariableWL" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves unconditional $`n^{\\Omega(k)}`$ deterministic time lower bounds for joint and separate k-dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed k in the specified sequential adjacency-matrix models", "coveredBy": [ "WeisfeilerLeman" ], "note": "MainClaim: ∃ c>0 ∃ k₀ ∀ k ≥ k₀ ∀ convention (joint/separate) ∀ input class (unrestricted / both graphs diameter ≤ 2 with n>0) ∀ model A (multitape TM, or log-word RAM whose word operations are TM-computable in poly(word length) time), if A decides k-WL equivalence on adjacency-matrix encodings of same-order pairs in the class, then ∃ n₀ ∀ n ≥ n₀, n^{ck} ≤ worst-case time at order n. No complexity assumption." }, { "clause": "With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs", "coveredBy": [ "VariableWL" ], "note": "EXPTIMEComplete WL ∧ EXPTIMEComplete SubWL, with k ≥ 2 in binary after two explicit adjacency matrices; SubWL restricts to equal positive order, connected, max degree ≤ 3. Joint-update colors. Hardness is under polynomial-time many-one reductions computed by a TM." }, { "clause": "deciding whether refinement identifies a graph is also EXPTIME-complete", "coveredBy": [ "WLIdentification" ], "note": "EXPTIMEComplete dimensionLanguage, where the language is {(G, k) : G nonempty simple, k > 0 in binary without leading zeros, WLdim G ≤ k}; Identifies quantifies over every comparison graph H. Because k-WL power is monotone in k (k = 1 is colour refinement, which 2-tuple joint refinement simulates), WLdim G ≤ k ⇔ k-WL identifies G." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "134", "title": "Generalized star height at most three", "headline": "Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem.", "challenges": [ "GeneralizedStarHeight" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet.", "coveredBy": [ "GeneralizedStarHeight" ], "note": "Stated directly." }, { "clause": "This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem.", "coveredBy": [ "GeneralizedStarHeight" ], "note": "The bound 3 is a constant in the statement." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch1.json" ] }, { "family": "135", "title": "Homogeneous depth-five lower bounds for iterated matrix multiplication", "headline": "Over every characteristic-zero field, the $`(1,1)`$ entry of a product of n independent $`n\\times n`$ variable matrices requires $`n^{\\Theta(\\sqrt n)}`$ gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables.", "challenges": [ "DepthFive" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Over every characteristic-zero field, the $`(1,1)`$ entry of a product of n independent $`n\\times n`$ variable matrices requires $`n^{\\Theta(\\sqrt n)}`$ gates in homogeneous depth-five sum–product circuits.", "coveredBy": [ "DepthFive" ], "note": "lower bound n^(sqrt n/400) for n >= threshold, threshold uniform over all char-0 fields (conjunct 2; conjunct 1 is ℂ); upper bound n^(sqrt n+4) over every field for n>=2 (conjunct 3)" }, { "clause": "This sharp bound allows shared gates and bottom linear forms involving all variables.", "coveredBy": [ "DepthFive" ], "note": "gates are indexed and reused by any number of parents; bottom Σ gates are arbitrary linear combinations of leaves" } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch8.json" ] }, { "family": "137", "title": "One-tape time simulation in two-fifths-power space", "headline": "Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time T using $`O(T^{2/5}\\log^C(T+2))`$ space, improving the square-root exponent. Heads move at most one cell per step; finitely many read-only input heads are allowed. Initial contents are independent of T, and contents and input symbols have polylogarithmic-space access. Simulation time is unrestricted.", "challenges": [ "OneTapeSpace" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time T using $`O(T^{2/5}\\log^C(T+2))`$ space", "coveredBy": [ "OneTapeSpace" ], "note": "capped: for the fixed source machine M (variable, quantified before everything) and any accessor A with constants p, q ≥ 1, ∃ a fixed simulator TM N and e ∈ ℕ, c > 0 such that for every input x and every T ≥ 2 (given in binary) for which the accessor works, N halts with output M.outcome x T = (state at time T, halted flag) inside a work-tape window of ≤ c·T^{2/5}·(log₂(T+2))^e bits. Halted states are frozen by Single.move, so 'halted at T' = 'halted by T'." }, { "clause": "improving the square-root exponent", "coveredBy": [ "OneTapeSpace" ], "note": "Context: the stated exponent is 2/5 (Cost.exponent)." }, { "clause": "Heads move at most one cell per step; finitely many read-only input heads are allowed", "coveredBy": [ "OneTapeSpace" ], "note": "Dir ∈ {L,N,R} with delta ∈ {−1,0,1}; Single has heads : ℕ read-only heads reading sensor x (some i)." }, { "clause": "Initial contents are independent of T, and contents and input symbols have polylogarithmic-space access", "coveredBy": [ "OneTapeSpace" ], "note": "sensor : List X → Option (Fin heads) → ℤ → A has no T argument; Access A p q x T requires a TM computing every symbol at distance ≤ T within p·(Log T)^q work bits." }, { "clause": "Simulation time is unrestricted", "coveredBy": [ "OneTapeSpace" ], "note": "Within bounds only the window (space), not the run length." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "139", "title": "Subpolynomial query complexity for log-concave sampling", "headline": "For C2 potentials with a supplied minimizer and $`I\\preceq\\nabla^2V\\preceq2I`$, proves that sampling within total variation 1/10 requires only $`C_\\varepsilon d^\\varepsilon`$ exact value-and-gradient queries for every fixed ε > 0. The bound holds on every run, with unrestricted computation between queries. A logarithmic lower bound also holds, so the optimal power-law exponent in this oracle model is zero.", "challenges": [ "LogConcaveQuery" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "For C2 potentials with a supplied minimizer and $`I\\preceq\\nabla^2V\\preceq2I`$, proves that sampling within total variation 1/10 requires only $`C_\\varepsilon d^\\varepsilon`$ exact value-and-gradient queries for every fixed ε > 0", "coveredBy": [ "LogConcaveQuery" ], "note": "First conjunct: ∀ ε>0 ∃ C ∀ d ≥ 2, queryComplexity d ≤ C·d^ε, where queryComplexity is the least q such that one randomized measurable adaptive algorithm making exactly q (V(x), ∇V(x)) queries has output law within TV 1/10 of the Gibbs law for EVERY admissible V (C², V(0)=0, ∇V(0)=0, I ≤ ∇²V ≤ 2I). Bounding a ℕ∞ by a finite real also forces queryComplexity < ⊤." }, { "clause": "The bound holds on every run, with unrestricted computation between queries", "coveredBy": [ "LogConcaveQuery" ], "note": "The query count q is a type-level constant of OracleAlgorithm (every execution makes q queries); query/output maps are arbitrary measurable functions of (randomness, transcript)." }, { "clause": "A logarithmic lower bound also holds, so the optimal power-law exponent in this oracle model is zero", "coveredBy": [ "LogConcaveQuery" ], "note": "Second conjunct: ∃ c>0 ∃ d₀ ≥ 2 ∀ d ≥ d₀, c·log d ≤ queryComplexity d; third: gammaStar = 0." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "140", "title": "Memory–sample lower bounds for noiseless Gaussian regression", "headline": "For fixed A > 0, a one-pass learner with $`Ad^2`$ persistent bits needs $`\\Omega_A(d\\log(1/\\epsilon))`$ noiseless Gaussian samples to recover a unit vector to angular error $`0\\lt \\epsilon\\le1/10`$ with probability 2/3, uniformly in accuracy for large d. Computation and randomized updates are unrestricted, but output uses only the terminal state, stopping index and fresh randomness.", "challenges": [ "MemoryPrecision", "NoiselessRegression", "PosteriorReplicas", "GaussianFiniteEntropy", "GaussianInformation", "ProjectionMoments", "GaussianReplacement", "SubsphereCurrent" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "For fixed A > 0, a one-pass learner with $`Ad^2`$ persistent bits needs $`\\Omega_A(d\\log(1/\\epsilon))`$ noiseless Gaussian samples to recover a unit vector to angular error $`0\\lt \\epsilon\\le1/10`$ with probability 2/3, uniformly in accuracy for large d.", "coveredBy": [ "NoiselessRegression", "ProjectionMoments", "MemoryPrecision" ], "note": "NoiselessRegression.FixedQuadraticMemory: ∀ A₀ > 0 ∃ c > 0 ∃ d₀ ∀ d ≥ d₀ ∀ M ≤ A₀d², T, ∀ ε ∈ (0, 1/10], every admissible learner with uniform-prior success ≥ 2/3 has c·d·log(1/ε) ≤ T. c and d₀ precede ε, so the bound is uniform in accuracy. ProjectionMoments repeats it for kernel learners with seeds." }, { "clause": "Computation and randomized updates are unrestricted, but output uses only the terminal state, stopping index and fresh randomness.", "coveredBy": [ "NoiselessRegression" ], "note": "Learner: initial choice, transitions and output are arbitrary (a.e.-measurable) functions of a shared seed ω, the time index and the 2^M-valued state; the estimate is output(stop index, ω, terminal state). Output may use the shared seed, which is more permissive than 'fresh randomness', so the lower bound covers the headline's learners." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": "Second reader: confirm the elaboration of ((2 : ℕ) / (3 : ℕ) : ℝ≥0∞) in NoiselessRegression.lean:105,106,115 (expected 2/3 in ℝ≥0∞; a #check against the pinned Mathlib would settle it). Either way it cannot make the statement cheaper.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "NoiselessRegression", "flag": "JUNK? (checked, benign): the threshold is written ((2 : ℕ) / (3 : ℕ) : ℝ≥0∞) (lines 105, 106, 115). Lean's binop% elaborator propagates the expected type ℝ≥0∞ to the leaves (the Nat.cast_div idiom), so this is ↑2/↑3 = 2/3 in ℝ≥0∞, not ℕ-division 0. Even if it were 0, the statement would become false (a T = 0 learner meets it), not weaker." }, { "challenge": "MemoryPrecision", "flag": "JUNK? (checked, benign): InverseGram uses the real inverse of √det, so it is 0 rather than ∞ on degenerate frames (line 161); the same conjunct asserts a.e. linear independence (lines 516-518), so the junk set is null." }, { "challenge": "MemoryPrecision", "flag": "JUNK? (checked, benign): PointwiseDensity returns 0 when the δ → 0 limit fails to exist (lines 274-278); PointwiseDensityProperties then asserts it is a density of the projection law (lines 664-665), so the default cannot carry the claim." }, { "challenge": "GaussianInformation", "flag": "Local instances ambientSub and ambientInner (lines 117-124) redefine Sub and Inner on EuclideanSpace for the AllScaleInverseVolume section; both are the standard coordinate formulas, so content is unchanged." }, { "challenge": "GaussianInformation", "flag": "JUNK? (checked, benign): B ν = ∫ log((J ν s).toReal) (line 156) would be a junk 0 if log J were not integrable, which would make the coercivity inequality trivial; under BoundedDensity on the sphere J is bounded above (exponent 2·(d/16) < d−1) and below (‖u−s‖ ≤ 2), so log J is bounded and integrable." }, { "challenge": "ProjectionMoments", "flag": "Mild: seeded learners need only a.e. CompletedRules (AEMeasurable transition probabilities); the success integrands are lintegrals whose measurability is not separately required (lines 228-231, 262-264). For a non-measurable integrand ∫⁻ is the lower integral, which makes the 2/3 hypothesis harder to meet — narrower only on pathological learners, and NoiselessRegression carries the headline anyway." }, { "challenge": "GaussianReplacement", "flag": "JUNK? (checked, benign): shannonEntropy (lines 93-94) is 0 for a non-atomic law (log 0 = 0); every use has W Countable or Fintype with MeasurableSingletonClass, so the laws are atomic." }, { "challenge": "GaussianReplacement", "flag": "JUNK? (checked, benign): exactDensity defaults to 0 when the cube-average limit fails (lines 37-41); the statements assert the limit exists a.e. (actual_limit) before using it." }, { "challenge": "GaussianReplacement", "flag": "JUNK? (checked, benign): averagedFreshInformation uses .toReal of conditional mutual information (line 316); with finite message sets that information is ≤ log|V| < ∞, so toReal is not truncating ∞ to 0." } ], "flagReadings": [ { "challenge": "NoiselessRegression", "flag": "JUNK? (checked, benign): the threshold is written ((2 : ℕ) / (3 : ℕ) : ℝ≥0∞) (lines 105, 106, 115). Lean's binop% elaborator propagates the expected type ℝ≥0∞ to the leaves (the Nat.cast_div idiom), so this is ↑2/↑3 = 2/3 in ℝ≥0∞, not ℕ-division 0. Even if it were 0, the statement would become false (a T = 0 learner meets it), not weaker.", "decision": "cleared", "why": "The type ascription elaborates the binop% tree against the expected type ℝ≥0∞, which seeds the tree's max type, so the coercion goes in at the leaves: ↑(2:ℕ) / ↑(3:ℕ) = 2/3 in ℝ≥0∞, not ↑((2:ℕ)/(3:ℕ)) = ↑0. This is the same rule that makes (n / m : ℝ) real division. The solution tree corroborates it: Main.lean:38 passes hs (carrying this threshold) directly as the argument hs : (2/3 : ℝ≥0∞) ≤ … of streaming_success_bound_real (Precision.lean:26). That typechecks only if the two thresholds are definitionally equal: ↑(2:ℕ) unfolds to the OfNat 2 literal of ℝ≥0∞, and ↑0 could not. Had it been 0, the success hypothesis would be vacuous and the conclusion c·d·log(1/ε) ≤ T would be demanded even of a T = 0 learner, which is false, not cheaper. Caveats: nothing was executed (no #check against the pinned Mathlib), and lane K has not run NoiselessRegression.", "evidence": [ "lean/ComparatorChallenges/NoiselessRegression.lean:105: (((2 : ℕ) / (3 : ℕ) : ℝ≥0∞) ≤ L.uniformSuccess ρ ε ∨", "lean/ComparatorChallenges/NoiselessRegression.lean:115: ((2 : ℕ) / (3 : ℕ) : ℝ≥0∞) ≤ L.uniformSuccess ρ ε →", "lean/OAI/Probability/GaussianRegression/Main.lean:38: (L.streaming_success_bound_real ρ hL (by omega) hε ha hq hqa hs)", "lean/OAI/Probability/GaussianRegression/Precision.lean:26: (hs : (2/3 : ℝ≥0∞) ≤ L.uniformSuccess ρ ε) :" ] }, { "challenge": "MemoryPrecision", "flag": "JUNK? (checked, benign): InverseGram uses the real inverse of √det, so it is 0 rather than ∞ on degenerate frames (line 161); the same conjunct asserts a.e. linear independence (lines 516-518), so the junk set is null.", "decision": "cleared", "why": "InverseGram is 0 exactly where the Gram determinant of (v.2 i − v.1) vanishes (the statement has 1 ≤ m, so the power does not mask it). In a real inner product space that is exactly where those vectors are linearly dependent. AmbientGramStatement asserts linear independence μ⊗μ^(p−1)-a.e. as the conjunct immediately before the integral bound (516-518). So the junk set is null, and the lintegral at 519 equals the integral of the genuine det^(−m/2): the bound is about the true quantity.", "evidence": [ "lean/ComparatorChallenges/MemoryPrecision.lean:161: ENNReal.ofReal ((Real.sqrt (Matrix.gram ℝ (fun i => v.2 i - v.1)).det)⁻¹ ^ m)", "lean/ComparatorChallenges/MemoryPrecision.lean:516: (∀ᵐ v : Space d × (Fin (p-1) → Space d)", "lean/ComparatorChallenges/MemoryPrecision.lean:518: LinearIndependent ℝ (fun i => v.2 i - v.1)) ∧", "lean/ComparatorChallenges/MemoryPrecision.lean:519: (∫⁻ v : Space d × (Fin (p-1) → Space d), InverseGram m v" ] }, { "challenge": "MemoryPrecision", "flag": "JUNK? (checked, benign): PointwiseDensity returns 0 when the δ → 0 limit fails to exist (lines 274-278); PointwiseDensityProperties then asserts it is a density of the projection law (lines 664-665), so the default cannot carry the claim.", "decision": "cleared", "why": "The default 0 is taken only where the δ = 1/i limit fails. PointwiseDensityProperties asserts that ofReal ∘ PointwiseDensity is a density of ProjectionLaw μ m against the reference (664-665), and that any other measurable density equals it reference-a.e. (669), so PointwiseDensity is the true density a.e. The only conjunct a junk 0 could make cheaper is the L^p bound (670). That integral sees PointwiseDensity only up to reference-null sets, so it is the L^p norm of the true density. FullRankDensityFormula is a pointwise-in-A equality of measures, which a wrong value could only falsify.", "evidence": [ "lean/ComparatorChallenges/MemoryPrecision.lean:277: exact if h : ∃ l : ℝ, Tendsto (fun i : ℕ => GaussianRegularization μ m (1/(i:ℝ)) x)", "lean/ComparatorChallenges/MemoryPrecision.lean:278: atTop (𝓝 l) then Classical.choose h else 0", "lean/ComparatorChallenges/MemoryPrecision.lean:664: reference.withDensity (fun x => ENNReal.ofReal (PointwiseDensity μ m x)) =", "lean/ComparatorChallenges/MemoryPrecision.lean:669: ProjectionLaw μ m → PointwiseDensity μ m =ᵐ[reference] F) ∧", "lean/ComparatorChallenges/MemoryPrecision.lean:670: (∫⁻ x, ENNReal.ofReal (PointwiseDensity μ m x)^p ∂reference) ≤" ] }, { "challenge": "GaussianInformation", "flag": "JUNK? (checked, benign): B ν = ∫ log((J ν s).toReal) (line 156) would be a junk 0 if log J were not integrable, which would make the coercivity inequality trivial; under BoundedDensity on the sphere J is bounded above (exponent 2·(d/16) < d−1) and below (‖u−s‖ ≤ 2), so log J is bounded and integrable.", "decision": "cleared", "why": "The mechanism is real in principle: if log J were not ν-integrable, then B ν = 0 and (1/2)KL − Cd ≤ KL − B would hold trivially. It cannot occur under BoundedDensity. Since ν ≤ H·σ and σ = sphereProbability is the radial pushforward of the uniform ball law (the uniform law on the unit sphere, with {0} null), ν-a.e. u and s lie on the sphere. There ‖u−s‖ ≤ 2 gives J ν s ≥ 2^(−k) with k = 2·(d/16). In the other direction, J ν s ≤ H·∫‖u−s‖^(−k) dσ(u) < ∞, because k ≤ d/8 < d−1 = dim S^(d−1). So log J is ν-a.e. bounded, hence integrable, and B is the true ∫ log J. In addition, KL(ν‖σ) ≤ log H < ∞, so its toReal does not truncate ∞. The other inputs of Phi are also bounded-density laws: σ itself, finite mixtures, and the posteriors of a law whose first marginal is σ. Lane K has not run GaussianInformation.", "evidence": [ "lean/ComparatorChallenges/GaussianInformation.lean:154: ∫⁻ u,((ENNReal.ofReal ‖u-s‖)⁻¹)^(2*(d/16)) ∂ν", "lean/ComparatorChallenges/GaussianInformation.lean:156: ∫ s, Real.log ((J ν s).toReal) ∂ν", "lean/ComparatorChallenges/GaussianInformation.lean:161: ∃ H : ℝ, 0 < H ∧ ν ≤ ENNReal.ofReal H • sphereProbability d", "lean/ComparatorChallenges/GaussianInformation.lean:186: (∀ ν : Measure (Ambient d), IsProbabilityMeasure ν → BoundedDensity ν →" ] }, { "challenge": "GaussianReplacement", "flag": "JUNK? (checked, benign): shannonEntropy (lines 93-94) is 0 for a non-atomic law (log 0 = 0); every use has W Countable or Fintype with MeasurableSingletonClass, so the laws are atomic.", "decision": "cleared", "why": "Every message type that feeds shannonEntropy is Countable (CriticalComparisonMain, actualRow*) or Finite/Fintype (OneLevel, TwoLabel, HybridBlock), always with MeasurableSingletonClass. On such a type every finite measure is purely atomic, π = Σ π{w} δ_w, so the lintegral is Σ_w π{w}·(−log π{w}), the true Shannon entropy. Points with π{w} = 0 carry no mass, so the log 0 = 0 junk there is integrated against nothing. The laws are probability measures (maps and marginals of a probability P), so π{w} ≤ 1 and ofReal never clips. Wherever .toReal is taken the entropy is finite: by hEntropy : ≠ ∞ (actualRow*), by Finite U (OneLevel), or by the ≤ ofReal d² / ≤ ofReal B hypotheses.", "evidence": [ "lean/ComparatorChallenges/GaussianReplacement.lean:94: ∫⁻ w, ENNReal.ofReal (-Real.log (π {w}).toReal) ∂π", "lean/ComparatorChallenges/GaussianReplacement.lean:239: ∀ (W : Type) [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W],", "lean/ComparatorChallenges/GaussianReplacement.lean:248: ∀ (U : Type) [MeasurableSpace U] [Finite U] [MeasurableSingletonClass U],", "lean/ComparatorChallenges/GaussianReplacement.lean:437: [Countable W] [MeasurableSingletonClass W]", "lean/ComparatorChallenges/GaussianReplacement.lean:443: (hEntropy : shannonEntropy (P.map swapLast).snd ≠ ∞)" ] }, { "challenge": "GaussianReplacement", "flag": "JUNK? (checked, benign): exactDensity defaults to 0 when the cube-average limit fails (lines 37-41); the statements assert the limit exists a.e. (actual_limit) before using it.", "decision": "cleared", "why": "There are three uses, and none reaches the default 0 on positive mass. (1) MixedMomentMain (59-61, 66) asserts that, off a ρ-null set of s and for a.e. G, the cube average converges along 𝓝[>]0 to exactDensity. cubeAverage is finite (ρ is finite). If the 1/j toReal limit failed, exactDensity would be 0, and the asserted convergence to 0 would itself supply that limit, a contradiction. The positivity conjunct (66) excludes the default as well. (2) The actual_limit field of ActualRow(Source)Conclusion (156-158) pins exactDensity to the a.e. limit before exact_bound and extended_log_bound read it. This matters: a junk 0 on positive mass would make logNegative = ∞, hence extendedLogExpectation = ⊥ and extended_log_bound trivial. (3) In fiberTwoPointLeft (380) exactDensity is a density inside an equality of measures. By Lebesgue differentiation the cube-average limit exists at Lebesgue-a.e. label point, and for a.e. G (k < d−1) the label of a uniform s has an absolutely continuous law, so the default is taken only on a null set (and a wrong density could only falsify the equality).", "evidence": [ "lean/ComparatorChallenges/GaussianReplacement.lean:41: exact if h : ∃ L : ℝ, Tendsto f atTop (𝓝 L) then ENNReal.ofReal h.choose else 0", "lean/ComparatorChallenges/GaussianReplacement.lean:61: (𝓝[>] 0) (𝓝 (exactDensity ρ G (label G (s : Vector d))))) ∧", "lean/ComparatorChallenges/GaussianReplacement.lean:66: 0 < exactDensity ρ G (label G (s : Vector d)) ∧", "lean/ComparatorChallenges/GaussianReplacement.lean:156: actual_limit : ∀ᵐ p ∂actualExperiment (r:=r) P,", "lean/ComparatorChallenges/GaussianReplacement.lean:158: (𝓝 (selectedExact ρ (actualObservation p)))", "lean/ComparatorChallenges/GaussianReplacement.lean:380: (fun z => exactDensity (uniformSphere d) z.1 (label z.1 (z.2:Vector d)))) ⊗ₘ" ] }, { "challenge": "GaussianReplacement", "flag": "JUNK? (checked, benign): averagedFreshInformation uses .toReal of conditional mutual information (line 316); with finite message sets that information is ≤ log|V| < ∞, so toReal is not truncating ∞ to 0.", "decision": "cleared", "why": "averagedFreshInformation is used only in HybridBlockMain, where U and V are Fintype with MeasurableSingletonClass. freshInformation P k is the conditional mutual information I(S; U | G, labels), a KL against the product of the conditional kernels. For a finite message alphabet it is ≤ H(U | G, labels) ≤ log|U| < ∞, so .toReal does not truncate ∞ to 0. The divisor hybridCount d = … + 1 is ≥ 1.", "evidence": [ "lean/ComparatorChallenges/GaussianReplacement.lean:305: exposedInformation (P.prod (gaussianRows k d))", "lean/ComparatorChallenges/GaussianReplacement.lean:316: (∑ k ∈ Finset.Icc (hybridFirst d) (hybridLast d), (freshInformation P k).toReal) /", "lean/ComparatorChallenges/GaussianReplacement.lean:322: ∀ (U V : Type) [Fintype U] [Fintype V]", "lean/ComparatorChallenges/GaussianReplacement.lean:324: [MeasurableSingletonClass U] [MeasurableSingletonClass V]," ] } ], "readings": [ "first/batch3.json", "flags/flags1.json" ] }, { "family": "143", "title": "Hilbert's sixteenth problem: uniform bounds for limit cycles", "headline": "Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles.", "challenges": [ "QuinticLienard" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree.", "status": "unlinked", "coveredBy": [], "manuscript": "uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026", "evidence": [ "CONTENTS.md:3443: Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles. ([Lean](lean/docs/143.md))", "CONTENTS.md:3452: For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree.", "lean/docs/143.md:3: The following describes the scope of the Lean formalization related to the following accompanying paper(s):", "lean/docs/143.md:5: - [Two limit cycles for quintic Liénard systems](../../preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf)", "lean/docs/143.md:1: # Hilbert's sixteenth problem: uniform bounds for limit cycles", "lean/formalization.yaml:844: id: ../preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf", "preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/build/sections/introduction.tex:28: The problem is a restricted instance of the second part of", "lean/ComparatorChallenges/QuinticLienard.lean:10: (z.2 - F.eval z.1, -z.1)", "lean/ComparatorChallenges/QuinticLienard.lean:40: (∀ F : Polynomial ℝ, F.degree ≤ 5 → {C : Set Plane | IsLimitCycle F C}.encard ≤ 2) ∧" ], "why": "No challenge states it: of the 416 statements only QuinticLienard.lean defines periodic orbits or limit cycles, and only for x'=y−F(x), y'=−x with deg F ≤ 5; that theorem does not imply a bound for general planar polynomial fields, and no Lean declaration in the solution library's LienardCycles tree does either. The clause's only source is the uniform-bounds manuscript; the linked Liénard paper calls its own problem 'a restricted instance' and claims no uniform bound. The scope note links only the Liénard paper (formalization.yaml likewise lists only it). The note's title names the uniform bound, but under the rules a title neither declares an omission nor links a paper." }, { "clause": "For classical quintic Liénard systems, the exact maximum is two limit cycles.", "status": "stated", "coveredBy": [ "QuinticLienard" ], "evidence": [ "lean/ComparatorChallenges/QuinticLienard.lean:40: (∀ F : Polynomial ℝ, F.degree ≤ 5 → {C : Set Plane | IsLimitCycle F C}.encard ≤ 2) ∧", "lean/ComparatorChallenges/QuinticLienard.lean:41: (∃ F : Polynomial ℝ, F.degree ≤ 5 ∧ {C : Set Plane | IsLimitCycle F C}.encard = 2) := by" ], "encoding": [ "'quintic' = Polynomial.degree ≤ 5 (the manuscript's 'degree at most five'), lower degrees included", "limit cycle = range of a global nonconstant periodic solution with an open neighbourhood containing no other periodic-orbit image (isolated among periodic orbits; centres excluded)", "count = Set.encard of the set of images (geometrically distinct orbits)" ] } ], "gap": "The headline's main clause — the Hilbert's-sixteenth uniform bound on limit cycles of every real planar polynomial vector field by a constant depending only on the degree — is not stated; only the quintic Liénard count is.", "scopeNoteOnGap": "silent — the note's title names the uniform bound, its body describes only Liénard: lean/docs/143.md:1: # Hilbert's sixteenth problem: uniform bounds for limit cycles / lean/docs/143.md:9: For the quintic Liénard system $x'=y-F(x)$, $y'=-x$, the formalized result proves that every real polynomial $F$ of degree at most five yields at most two limit cycles, and that some such $F$ yields exactly two.", "note": "Read with particular care, the first reader's finding holds: the headline's lead claim, a resolution of the uniform-boundedness part of Hilbert's sixteenth problem, is stated by no challenge in the release, and only the quintic Liénard count is. The difference is the word, not the finding: the clause comes only from the uniform-bounds manuscript, which the note does not link, so under amendment 3 it is UNLINKED (the first reader had no UNLINKED). The UNLINKED should not be read as benign: the note's own title, and the CONTENTS.md ([Lean]) link placed right after the uniform-bound sentence, put the unformalized claim at the head of the Lean scope page, and no sentence of the note says it is outside. Both manuscripts were in the release's initial commit, dated the same day, so this is not a late addition.", "firstSuspicious": "Most consequential gap in the batch: the family title and headline announce Hilbert's sixteenth problem (uniform bounds), the September 24 uniform-bounds manuscript is not linked from the scope note, and no challenge states it.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json", "second/second4.json" ] }, { "family": "144", "title": "Banach’s simple Lebesgue-spectrum problem", "headline": "Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L2 space: the bilateral iterates of one real observable form an orthonormal basis of that space.", "challenges": [ "ThreeTorus" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics", "coveredBy": [ "ThreeTorus" ], "note": "Existence of a smooth measure-preserving T with simple Lebesgue spectrum on the mean-zero L2 space." }, { "clause": "A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L2 space", "coveredBy": [ "ThreeTorus" ], "note": "There exists T in Perm((R/Z)^3), smooth with smooth inverse (local C-infinity lifts), preserving the product Haar measure." }, { "clause": "the bilateral iterates of one real observable form an orthonormal basis of that space", "coveredBy": [ "ThreeTorus" ], "note": "f in H0 real a.e., v n = f o T^n a.e. for n in Z, v orthonormal, span dense in H0 = the complex mean-zero subspace of L2." } ], "gap": null, "scopeNoteOnGap": "no gap", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "145", "title": "Rokhlin’s multiple-mixing problem", "headline": "Proves that every invertible mixing probability-preserving transformation is mixing of all finite orders, resolving Rokhlin's multiple-mixing problem for a single transformation. Correlations among any finite collection of measurable sets converge to the product of their measures whenever all pairwise time separations diverge.", "challenges": [ "Rokhlin" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "every invertible mixing probability-preserving transformation is mixing of all finite orders", "coveredBy": [ "Rokhlin" ], "note": "For T a measurable equivalence preserving a probability μ and mixing (two-sided, |n| → ∞): mixing of every order k ≥ 3." }, { "clause": "Correlations among any finite collection of measurable sets converge to the product of their measures whenever all pairwise time separations diverge.", "coveredBy": [ "Rokhlin" ], "note": "Times 0 = t₀ < t₁ < … built from k−1 positive gaps; atTop on (N+)^{k−1} = every gap → ∞, which for increasing times is every pairwise separation → ∞." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": "Definition holes on both the hypothesis and the conclusion: MATCHES is conditional on lane K.", "confidence": "high for the reading of the bodies as written; the word stands only if lane K confirms the four definition bodies", "conditionalOnLaneK": [ { "challenge": "Rokhlin", "strict": "SAME BODIES" } ], "flags": [ { "challenge": "Rokhlin", "flag": "DEFINITION HOLE: Rokhlin.json definition_names = OAI.Rokhlin.timeMap, OAI.Rokhlin.IsMixing, OAI.Rokhlin.layoutTime, OAI.Rokhlin.MixingOfOrder — the hypothesis (IsMixing) and the conclusion (MixingOfOrder) are holes; Comparator checks only their types, so bodies like MixingOfOrder := True or IsMixing := False would pass. The reading is of the challenge's bodies; lane K must confirm the solution's bodies are these." } ], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "146", "title": "Positive metric entropy for the standard map", "headline": "Proves that the standard sine map on the two-dimensional torus has positive metric entropy with respect to area for every sufficiently large positive parameter. This establishes Sinai's positive-parameter-measure conjecture for the original family, with the stronger conclusion of a full parameter tail.", "challenges": [ "StandardMapComponents", "StandardMapEntropy", "StandardMapLyapunov" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that the standard sine map on the two-dimensional torus has positive metric entropy with respect to area for every sufficiently large positive parameter.", "coveredBy": [ "StandardMapEntropy", "StandardMapLyapunov" ], "note": "∃ k₀ > 0 ∀ k ≥ k₀, 0 < metricEntropy area (standardMap k)." }, { "clause": "This establishes Sinai's positive-parameter-measure conjecture for the original family, with the stronger conclusion of a full parameter tail.", "coveredBy": [ "StandardMapEntropy" ], "note": "A full tail [k₀, ∞) of parameters, hence a positive-measure parameter set." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": "StandardMapComponents.lean:27 contains a run_cmd that modifies an environment extension inside a Comparator statement file (supporting challenge only).", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "StandardMapComponents", "flag": "The challenge file executes a metaprogram: line 27 'run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}' (resets the auxiliary-lemma cache). It does not alter any stated proposition, but a command that edits environment state inside a challenge file is worth a look by lane K." } ], "flagReadings": null, "readings": [ "first/batch3.json" ] }, { "family": "148", "title": "The entropy-rate dimension formula for self-similar measures", "headline": "For every self-similar measure on the line generated by finitely many contracting similarities, proves $`\\dim_{\\mathrm H}\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}`$, where $`h_{\\mathrm{RW}}`$ is the entropy rate of random composed maps and χ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios.", "challenges": [ "SelfSimilar", "SelfSimilarCorollaries" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "For every self-similar measure on the line generated by finitely many contracting similarities, proves $`\\dim_{\\mathrm H}\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}`$, where $`h_{\\mathrm{RW}}`$ is the entropy rate of random composed maps and χ the average logarithmic contraction", "coveredBy": [ "SelfSimilar" ], "note": "For every finite IFS of similarities x -> r_i x + t_i (0 < |r_i| < 1, weights p_i > 0 summing to 1) and every probability measure mu with mu = sum p_i (S_i)_* mu: lowerHausdorffDimension mu = min(1, h_RW/chi). Here h_RW = inf_n H_{n+1}/(n+1) of the entropy (in bits) of the law of the composed affine map, and chi = -sum p_i log2 |r_i|." }, { "clause": "This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios", "coveredBy": [ "SelfSimilar" ], "note": "No separation hypothesis; signed, unequal ratios; maps compared as affine maps, so exact overlaps merge in the entropy." } ], "gap": null, "scopeNoteOnGap": "no gap", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "149", "title": "Classwise permanence for weakly reversible mass-action systems", "headline": "Proves the permanence conjecture for every finite weakly reversible mass-action system with fixed positive rate constants. Every positive stoichiometric compatibility class, even an unbounded one, has a common compact convex forward-invariant absorbing set. All positive trajectories in that class therefore eventually share positive lower and finite upper concentration bounds.", "challenges": [ "MassAction" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves the permanence conjecture for every finite weakly reversible mass-action system with fixed positive rate constants.", "status": "unlinked", "coveredBy": [], "manuscript": "Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026", "evidence": [ "lean/ComparatorChallenges/MassAction.lean:46: theorem global_bounded_persistent_solution :", "lean/ComparatorChallenges/MassAction.lean:49: ∀ x0 : Fin d → ℝ, PositiveState x0 →", "lean/ComparatorChallenges/MassAction.lean:50: ∃ ε : ℝ, 0 < ε ∧ ε < 1 ∧", "CONTENTS.md:3566: We prove the permanence conjecture for finite weakly reversible mass-action systems with fixed positive reaction rates.", "CONTENTS.md:3564: [Uniform Permanence in Weakly Reversible Mass-Action Systems](preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf)", "lean/docs/149.md:5: - [Boundedness and persistence of weakly reversible mass-action systems](../../preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf)", "lean/docs/149.md:9: The bound may depend on the network, rates, and initial state." ] }, { "clause": "Every positive stoichiometric compatibility class, even an unbounded one, has a common compact convex forward-invariant absorbing set.", "status": "unlinked", "coveredBy": [], "manuscript": "Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026", "evidence": [ "CONTENTS.md:3566: Every positive stoichiometric compatibility class admits one compact convex forward-invariant set that every positive trajectory in that class enters in finite time, even when the class is unbounded.", "CONTENTS.md:3564: [Uniform Permanence in Weakly Reversible Mass-Action Systems](preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf)", "lean/docs/149.md:5: - [Boundedness and persistence of weakly reversible mass-action systems](../../preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf)", "lean/docs/149.md:9: The bound may depend on the network, rates, and initial state." ] }, { "clause": "All positive trajectories in that class therefore eventually share positive lower and finite upper concentration bounds.", "status": "unlinked", "coveredBy": [], "manuscript": "Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026", "evidence": [ "lean/ComparatorChallenges/MassAction.lean:49: ∀ x0 : Fin d → ℝ, PositiveState x0 →", "lean/ComparatorChallenges/MassAction.lean:53: ∀ t : ℝ, 0 ≤ t → ∀ i : Fin d, ε ≤ x t i ∧ x t i ≤ ε⁻¹ := by", "CONTENTS.md:3566: Thus, after its entry time, every concentration satisfies the same positive lower and finite upper bounds throughout that class.", "CONTENTS.md:3564: [Uniform Permanence in Weakly Reversible Mass-Action Systems](preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf)", "lean/docs/149.md:5: - [Boundedness and persistence of weakly reversible mass-action systems](../../preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf)", "lean/docs/149.md:9: The bound may depend on the network, rates, and initial state." ] } ], "gap": "The headline claims classwise permanence (October 5 manuscript 'Uniform Permanence in Weakly Reversible Mass-Action Systems'); the Comparator statement gives only boundedness and persistence with initial-state-dependent ε (September 25 manuscript). The scope note states the initial-state dependence but never says permanence is omitted.", "scopeNoteOnGap": "The bound may depend on the network, rates, and initial state.", "note": "MassAction states boundedness and persistence with ∀ x0 ∃ ε (bounds depending on the initial state) — what the headline's classwise permanence implies, not the other way round, so NARROWER, not DIFFERENT. Every headline clause (one absorbing set per compatibility class, shared bounds) comes from the October-5 'Uniform Permanence' manuscript, which docs/149.md does not link (it links only the September-25 boundedness/persistence paper; the title 'Classwise permanence' alone declares nothing). First reader: UNDECLARED, before UNLINKED existed; same facts.", "firstSuspicious": "Borderline DECLARED/UNDECLARED. The scope note's body accurately describes the narrower statement ('The bound may depend on the network, rates, and initial state', 149.md:9) but never says that classwise permanence is outside the formalization; it links only the September boundedness/persistence paper (149.md:5), not the October permanence paper the headline summarizes, and its own title is 'Classwise permanence for weakly reversible mass-action systems' (149.md:1). Read as UNDECLARED because no omission is declared; a second reader should decide.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [ { "challenge": "MassAction", "flag": "NARROWER — UNDECLARED: permanence/absorbing set absent; scope note does not say it is omitted." }, { "challenge": "MassAction", "flag": "QUANTIFIER: ∀ x0 ∃ ε (MassAction.lean:49-50) versus the headline's one bound per compatibility class for all its positive trajectories." } ], "flagReadings": null, "readings": [ "first/batch3.json", "second/second5.json" ] }, { "family": "150", "title": "Weak mixing of triangular billiards with an irrational angle", "headline": "Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions.", "challenges": [ "IrrationalTriangleBilliard" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction.", "status": "unlinked", "coveredBy": [], "manuscript": "Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026", "evidence": [ "preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026/build/main.tex:31-34: We prove that the unit-speed billiard flow in every nondegenerate Euclidean ⏎ triangle with at least one angle irrational relative to \\(\\pi\\) is weakly ⏎ mixing for normalized Liouville measure. Equivalently, the product of the ⏎ flow with itself is ergodic.", "lean/docs/150.md:5: - [Ergodicity of triangular billiards with an irrational angle](../../preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026.pdf)", "lean/docs/150.md:9: The formalization proves that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to $\\pi$ is ergodic for normalized area times uniform angular measure. It constructs the flow outside the null set of exceptional trajectories, proves uniqueness of the flight chain there, and establishes measure preservation and the flow law almost everywhere. No genericity or Diophantine condition on the irrational angle is assumed.", "lean/ComparatorChallenges/IrrationalTriangleBilliard.lean:115-117: (∀ A : Set Phase, MeasurableSet A → ⏎ (∀ t : ℝ, phaseMeasure Q (((billiardFlow Q t) ⁻¹' A) ∆ A) = 0) → ⏎ phaseMeasure Q A = 0 ∨ phaseMeasure Q A = 1)" ] }, { "clause": "This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions.", "status": "stated", "coveredBy": [ "IrrationalTriangleBilliard" ], "partial": "stated as ergodicity on the whole class with no further hypothesis; the strengthening itself (weak mixing) is clause 1", "evidence": [ "lean/ComparatorChallenges/IrrationalTriangleBilliard.lean:41-42: def HasIrrationalAngle (Q : Triangle) : Prop := ⏎ ∃ i : Fin 3, Irrational (Q.angle i / Real.pi)", "lean/ComparatorChallenges/IrrationalTriangleBilliard.lean:119-120: theorem irrational_triangle_billiard (Q : Triangle) (hQ : Q.HasIrrationalAngle) : ⏎ MainConclusion Q := by" ] } ], "gap": "Weak mixing (from the October-5 manuscript, which the scope note does not link); only ergodicity is stated.", "scopeNoteOnGap": "silent", "note": "The challenge states ergodicity (invariant sets mod 0 have measure 0 or 1) with the flow built and shown measure-preserving; weak mixing, the headline claim, is the October-5 paper, which the note does not link (it links only the September-25 ergodicity paper; its title alone does not declare). Differs from the first reader only by amendment 3: the gap is UNLINKED, not UNDECLARED.", "firstSuspicious": "The note describes the narrower ergodicity statement accurately but never says weak mixing is left out.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "IrrationalTriangleBilliard", "flag": "NARROWER (undeclared): the last conjunct is ergodicity, not weak mixing" } ], "flagReadings": null, "readings": [ "first/batch1.json", "second/second6.json" ] }, { "family": "151", "title": "A C1 counterexample to the entropy conjecture", "headline": "Constructs a noninvertible C1 self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C1 self-maps: homological growth need not force positive orbit complexity.", "challenges": [ "C1EntropyCounterexample" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a noninvertible C1 self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology.", "coveredBy": [ "C1EntropyCounterexample" ], "note": "∃ q > 0 and f on S¹ × (S²)^(q+1), C¹ in the embedded sense, with coverEntropy f univ = 0, a nonzero v ∈ H₂(·;ℝ) with f_* v = 2v, and f not bijective." }, { "clause": "This disproves the homological entropy lower bound for general C1 self-maps: homological growth need not force positive orbit complexity.", "coveredBy": [ "C1EntropyCounterexample" ], "note": "Also states 0 < log 2 ≤ log(spectral radius on total real homology), so h_top = 0 < log(spectral radius)." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "C1EntropyCounterexample", "flag": "JUNK? (checked, benign): totalRealHomologySpectralRadius is an sSup (lines 55-63); on this compact manifold homology is finite-dimensional so the set is bounded; a junk 0 would make 'log 2 ≤ log …' false, not easier. The clause is redundant given the eigenvector." } ], "flagReadings": [ { "challenge": "C1EntropyCounterexample", "flag": "JUNK? (checked, benign): totalRealHomologySpectralRadius is an sSup (lines 55-63); on this compact manifold homology is finite-dimensional so the set is bounded; a junk 0 would make 'log 2 ≤ log …' false, not easier. The clause is redundant given the eigenvector.", "decision": "cleared", "why": "The set is {|a+ib| : a+ib an eigenvalue of f_* on H_n(M;ℝ) for some n}, because f_*(u+iv) = (a+ib)(u+iv) with u+iv ≠ 0. M = S¹ × (S²)^{q+1} is a compact (2q+3)-manifold, so each H_n is finite-dimensional and H_n = 0 for n > 2q+3. The set is therefore finite. It is nonempty because the theorem's own conjunct supplies v ≠ 0 with f_* v = 2v, giving r = 2 (a = 2, b = 0, u = v, v = 0). sSup is thus the true spectral radius of total homology, and it is ≥ 2. The junk value would anyway give log 0 = 0 < log 2, which makes the statement false, not easier.", "evidence": [ "lean/ComparatorChallenges/C1EntropyCounterexample.lean:57: sSup {r : ℝ |", "lean/ComparatorChallenges/C1EntropyCounterexample.lean:60: (u ≠ 0 ∨ v ≠ 0) ∧", "lean/ComparatorChallenges/C1EntropyCounterexample.lean:61: realHomologyMap n f hf u = a • u - b • v ∧", "lean/ComparatorChallenges/C1EntropyCounterexample.lean:62: realHomologyMap n f hf v = b • u + a • v ∧", "lean/ComparatorChallenges/C1EntropyCounterexample.lean:63: r = Real.sqrt (a ^ 2 + b ^ 2)}", "lean/ComparatorChallenges/C1EntropyCounterexample.lean:66: ∃ q : ℕ, 0 < q ∧ ∃ f : CounterexampleManifold q → CounterexampleManifold q, ∃ hf : IsC1SelfMap q f, Dynamics.coverEntropy f (Set.univ : Set (CounterexampleManifold q)) = (0 : EReal) ∧ (∃ v : RealSingularHomology 2 (CounterexampleManifold q), v ≠ 0 ∧ realHomologyMap 2 f hf.1 v = (2 : ℝ) • v) ∧ 0 < Real.log (2 : ℝ) ∧ Real.log (2 : ℝ) ≤ Real.log (totalRealHomologySpectralRadius f hf.1) ∧ ¬ Function.Bijective f ∧ 5 ≤ 2 * q + 3 := by" ] } ], "readings": [ "first/batch3.json", "flags/flags2.json" ] }, { "family": "152", "title": "Zero entropy does not guarantee a smooth positive-volume model", "headline": "Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension.", "challenges": [ "SmoothObstruction" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space", "coveredBy": [ "SmoothObstruction" ], "note": "Lean gives FiniteKSEntropy (all partition entropy rates exist and are bounded by one constant), not zero entropy." }, { "clause": "that is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension.", "coveredBy": [ "SmoothObstruction" ], "note": "¬ HasSmoothPositiveVolumeModel: no conull measurable conjugacy to a C^∞ diffeomorphism preserving a probability with smooth positive chart densities on any compact (boundaryless or with boundary) manifold of any dimension d." } ], "gap": "Zero Kolmogorov–Sinai entropy is claimed; the statement gives only finite entropy.", "scopeNoteOnGap": "The linked statement gives finite entropy; the paper's stronger zero-entropy conclusion is outside this statement.", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch3.json" ] }, { "family": "155", "title": "A counterexample to periodic tiling in dimension three", "headline": "Constructs a finite translational tile in ℤ3 that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in ℝ3, even with arbitrary real translation vectors.", "challenges": [ "PeriodicTilingThree" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a finite translational tile in ℤ3 that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension.", "coveredBy": [ "PeriodicTilingThree" ], "note": "∃ nonempty finite T ⊆ ℤ³ with some A such that T ⊕ A = ℤ³ (bijective sum map), and no such A is invariant under a finite-index subgroup; plus IsLeast {d | such a tile exists in ℤ^d} 3." }, { "clause": "Its unit-cube thickening gives the same counterexample in ℝ3, even with arbitrary real translation vectors.", "coveredBy": [ "PeriodicTilingThree" ], "note": "The thickening T + [0,1]³ tiles ℝ³ a.e. by some A ⊆ ℝ³, and no such A (arbitrary real translations) is invariant under a full-rank lattice." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch5.json" ] }, { "family": "156", "title": "Borsuk's conjecture fails in dimension nine", "headline": "Constructs a compact subset of ℝ9 that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in ℝ4, with the Frobenius metric.", "challenges": [ "BorsukNine" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a compact subset of ℝ9 that cannot be covered by ten sets of strictly smaller diameter", "coveredBy": [ "BorsukNine" ], "note": "projectorSet = {uuᵀ : ‖u‖ = 1, u ∈ ℝ⁴} in EuclideanSpace ℝ (Fin 4 × Fin 4) (Frobenius metric); the theorem states it is compact, lies in the trace-one symmetric matrices (a 9-dimensional affine subspace, isometric to ℝ⁹), has diameter √2, and admits no cover by ten subsets of diameter < √2." }, { "clause": "disproving Borsuk's covering assertion already in dimension nine", "coveredBy": [ "BorsukNine" ], "note": "Follows: d + 1 = 10 sets in dimension 9." }, { "clause": "The example consists of rank-one orthogonal projectors onto lines in ℝ4, with the Frobenius metric", "coveredBy": [ "BorsukNine" ], "note": "projector u = (u_i u_j)_{ij} with the ℓ² norm on Fin 4 × Fin 4 entries." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "157", "title": "Graph coloring, clique minors, and Colin de Verdière invariants", "headline": "Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy $`\\chi_f(G)\\gt h(G)`$, where $`h(G)`$ is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound $`\\chi_f(G)\\le\\mu(G)+1`$. In the positive direction, every finite nonempty graph satisfies $`\\chi_{\\mathrm{list}}(G)\\le C h(G)`$ for a universal constant C.", "challenges": [ "ListHadwiger", "HadwigerCounterexample" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy $`\\chi_f(G)\\gt h(G)`$, where $`h(G)`$ is the largest clique-minor order.", "coveredBy": [ "HadwigerCounterexample" ], "note": "The Lean states the ordinary chromatic form with explicit bounds: ∀ M ∃ m ≥ max M 5 and G on m vertices with α(G) ≤ 2, h(G) < 26m/75 + 2/3 < m/2 ≤ χ(G). χ_f is not defined, and the fractional form is declared out of scope. But it follows from what IS stated by the one-line bound χ_f ≥ |V|/α ≥ m/2 > h(G). Covered up to that classical step." }, { "clause": "Also disproves the fractional Colin de Verdière chromatic bound $`\\chi_f(G)\\le\\mu(G)+1`$.", "coveredBy": [], "note": "Not stated (μ is not defined); declared out of scope. Not implied: since μ ≥ h − 1, refuting χ_f ≤ μ + 1 needs an upper bound on μ that the Lean does not give." }, { "clause": "In the positive direction, every finite nonempty graph satisfies $`\\chi_{\\mathrm{list}}(G)\\le C h(G)`$ for a universal constant C.", "coveredBy": [ "ListHadwiger" ], "note": "∃ C ≥ 1 ∀ finite nonempty V ∀ G: χ_list(G) ≤ C·h(G); C is chosen before the graph." } ], "gap": "The Colin de Verdière clause is not stated. The fractional clause is not stated as such (no χ_f), though it follows from the stated α ≤ 2 and h < m/2.", "scopeNoteOnGap": "lean/docs/157.md:12: \"This disproves the ordinary chromatic form. The paper's fractional-chromatic strengthening and the Colin de Verdière consequence are outside this selected statement.\"", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "ListHadwiger", "flag": "JUNK? checked, none: both sInf (lists) and sSup (minors) are over nonempty bounded sets for finite nonempty V." } ], "flagReadings": [ { "challenge": "ListHadwiger", "flag": "JUNK? checked, none: both sInf (lists) and sSup (minors) are over nonempty bounded sets for finite nonempty V.", "decision": "cleared", "why": "Here one junk direction is dangerous: if {k | Choosable G k} were empty, listChromaticNumber = sInf ∅ = 0 and `0 ≤ C * h` would hold trivially. It is not empty: Choosable G (card V) holds by greedy colouring, since each vertex has at most card V − 1 neighbours and every list has at least card V colours; Choosable is monotone in k; and 0 is not in the set for nonempty V (empty lists admit no colouring), so the sInf is the true choice number ≥ 1. The other direction is harmless: a junk Hadwiger number (Nat sSup of an unbounded set is 0) would make the claim false, since the choice number is ≥ 1. It is not reached either: SimpleGraph.Connected includes Nonempty, so branch sets are nonempty and pairwise disjoint, t ≤ card V, and 0 and 1 are in the set. Both quantities are genuine. Lane K: CERTIFIED.", "evidence": [ "ComparatorChallenges/ListHadwiger.lean:11: def Choosable {V : Type} (G : SimpleGraph V) (k : ℕ) : Prop :=", "ComparatorChallenges/ListHadwiger.lean:12: ∀ (Color : Type) (L : V → Finset Color), (∀ v, k ≤ (L v).card) → ListColorable G L", "ComparatorChallenges/ListHadwiger.lean:14: noncomputable def listChromaticNumber {V : Type} (G : SimpleGraph V) : ℕ :=", "ComparatorChallenges/ListHadwiger.lean:15: sInf {k : ℕ | Choosable G k}", "ComparatorChallenges/ListHadwiger.lean:17: def HasCliqueMinor {V : Type} (G : SimpleGraph V) (t : ℕ) : Prop :=", "ComparatorChallenges/ListHadwiger.lean:18: ∃ B : Fin t → Set V,", "ComparatorChallenges/ListHadwiger.lean:19: (∀ i, (G.induce (B i)).Connected) ∧", "ComparatorChallenges/ListHadwiger.lean:20: (∀ i j, i ≠ j → Disjoint (B i) (B j)) ∧", "ComparatorChallenges/ListHadwiger.lean:21: (∀ i j, i ≠ j → ∃ v ∈ B i, ∃ w ∈ B j, G.Adj v w)", "ComparatorChallenges/ListHadwiger.lean:23: noncomputable def hadwigerNumber {V : Type} (G : SimpleGraph V) : ℕ :=", "ComparatorChallenges/ListHadwiger.lean:24: sSup {t : ℕ | HasCliqueMinor G t}", "ComparatorChallenges/ListHadwiger.lean:26: def MainStatement : Prop :=", "ComparatorChallenges/ListHadwiger.lean:27: ∃ C : ℕ, 1 ≤ C ∧ ∀ (V : Type) [Fintype V] [Nonempty V] (G : SimpleGraph V),", "ComparatorChallenges/ListHadwiger.lean:28: listChromaticNumber G ≤ C * hadwigerNumber G" ] } ], "readings": [ "first/batch5.json", "flags/flags3.json" ] }, { "family": "158", "title": "The Euclidean plane cannot be colored with five colors", "headline": "Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane.", "challenges": [ "EuclideanFiveColor", "PlaneColoring" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes.", "coveredBy": [ "EuclideanFiveColor" ], "note": "¬ ∃ coloring : ℂ → Fin 5 with ProperColoring 5 coloring (no measurability). This holds only if ProperColoring keeps its displayed body (see the DEFINITION HOLE flag)." }, { "clause": "This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane.", "coveredBy": [ "PlaneColoring", "EuclideanFiveColor" ], "note": "PlaneColoring: there is a proper 7-coloring of EuclideanSpace ℝ (Fin 2). With the five-color result, this pins the value to 6 or 7." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": "The MATCHES verdict assumes the displayed body of the definition hole ProperColoring. Until lane K's strict run (definition_names emptied) passes, the Comparator check alone does not prevent a solution from redefining it.", "confidence": "medium", "conditionalOnLaneK": [ { "challenge": "EuclideanFiveColor", "strict": "SAME BODIES" } ], "flags": [ { "challenge": "EuclideanFiveColor", "flag": "DEFINITION HOLE: OAI.EuclideanFiveColor.ProperColoring is in definition_names. Comparator compares only its name, universes and type (ℕ → (ℂ → Fin n) → Prop), never its body. A solution defining ProperColoring := fun _ _ => False would make the theorem trivially true. The whole conclusion is carried by this definition, so lane K's strict run must decide the body before this row stands." } ], "flagReadings": null, "readings": [ "first/batch5.json" ] }, { "family": "159", "title": "Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds", "headline": "Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed k ≥ 3, every subset of $`\\{1,\\ldots,N\\}`$ with no nonconstant k-term progression has size at most $`C_kN\\exp[-c_k(\\log N)^{\\varepsilon_k}]`$, with positive constants depending only on k.", "challenges": [ "ErdosReciprocal" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length.", "coveredBy": [ "ErdosReciprocal" ], "note": "∀ A ⊆ ℕ, ¬ Summable (n ↦ 1/n on A) ⇒ ∀ k, ∃ a, d > 0 with a + i·d ∈ A for i < k." }, { "clause": "Quantitatively, for each fixed k ≥ 3, every subset of $`\\{1,\\ldots,N\\}`$ with no nonconstant k-term progression has size at most $`C_kN\\exp[-c_k(\\log N)^{\\varepsilon_k}]`$, with positive constants depending only on k.", "coveredBy": [], "note": "The r_k(N) ≤ C_k N exp(−c_k (log N)^{ε_k}) bound is not stated; declared out." } ], "gap": "The quantitative Szemerédi bound for k-AP-free subsets of {1,…,N} is not stated.", "scopeNoteOnGap": "The selected theorem is the reciprocal-sum consequence. The paper's quantitative upper bound for the largest progression-free subset of $\\{1,\\ldots,N\\}$ is outside this statement.", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "ErdosReciprocal", "flag": "JUNK? (checked, benign): reciprocalTerm uses (0 : ℝ)⁻¹ = 0 at n = 0 (lines 16-18); a single term cannot affect summability." } ], "flagReadings": [ { "challenge": "ErdosReciprocal", "flag": "JUNK? (checked, benign): reciprocalTerm uses (0 : ℝ)⁻¹ = 0 at n = 0 (lines 16-18); a single term cannot affect summability.", "decision": "cleared", "why": "When 0 ∈ A, reciprocalTerm A 0 = (0:ℝ)⁻¹ = 0. Changing one term of a real family never changes Summable. All terms are nonnegative, so ¬Summable is exactly divergence of Σ_{n∈A, n≥1} 1/n, which is the prose hypothesis. Allowing 0 ∈ A also does not weaken the conclusion. For A ⊆ ℕ_{>0} (the prose's sets) the statement is literally the prose. When 0 ∈ A, ∀ k, HasAP A k still yields k-term progressions inside A∖{0}: drop the first term of a (k+1)-term progression with d > 0. The quantifier ∀ k makes the k = 0 vacuity irrelevant. (Lane K: not yet decided; the job stopped before Comparator. That has no bearing here.)", "evidence": [ "ComparatorChallenges/ErdosReciprocal.lean:7-8: def HasAP (A : Set ℕ) (k : ℕ) : Prop := ⏎ ∃ a d : ℕ, 0 < d ∧ ∀ i < k, a + i * d ∈ A", "ComparatorChallenges/ErdosReciprocal.lean:16-18: noncomputable def reciprocalTerm (A : Set ℕ) (n : ℕ) : ℝ := by ⏎ classical ⏎ exact if n ∈ A then (n : ℝ)⁻¹ else 0", "ComparatorChallenges/ErdosReciprocal.lean:24-25: def ReciprocalProgressionTheorem : Prop := ⏎ ∀ A : Set ℕ, ¬ Summable (reciprocalTerm A) → ∀ k : ℕ, HasAP A k" ] } ], "readings": [ "first/batch3.json", "flags/flags4.json" ] }, { "family": "160", "title": "Superexponential van der Waerden numbers", "headline": "Resolves Erdős's superexponential-growth question for van der Waerden numbers. If $`W_r(k)`$ is the least interval length forcing a monochromatic k-term progression in every r-coloring, then $`W_r(k)\\gt k^{ck\\lfloor\\log_2 r\\rfloor}`$ for an absolute c > 0, all r ≥ 2 and sufficiently large k, uniformly in r. In particular, $`W_r(k)^{1/k}\\to\\infty`$ for each fixed r.", "challenges": [ "QuantitativeVanDerWaerden" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves Erdős's superexponential-growth question for van der Waerden numbers.", "coveredBy": [ "QuantitativeVanDerWaerden" ], "note": "Via the bound below." }, { "clause": "If $`W_r(k)`$ is the least interval length forcing a monochromatic k-term progression in every r-coloring, then $`W_r(k)\\gt k^{ck\\lfloor\\log_2 r\\rfloor}`$ for an absolute c > 0, all r ≥ 2 and sufficiently large k, uniformly in r.", "coveredBy": [ "QuantitativeVanDerWaerden" ], "note": "Stated with explicit c = 1/100000 and quantifiers ∃K ∀k≥K ∀r≥2 (uniform in r)." }, { "clause": "In particular, $`W_r(k)^{1/k}\\to\\infty`$ for each fixed r.", "coveredBy": [ "QuantitativeVanDerWaerden" ], "note": "Implied, not stated: W^{1/k} > k^{c⌊log₂ r⌋} → ∞ since ⌊log₂ r⌋ ≥ 1." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "QuantitativeVanDerWaerden", "flag": "JUNK? (not triggered): W is sInf on ℕ, which is 0 for an empty set; the strict lower bound by a positive real is false at 0, so the statement itself forces the Ramsey set to be nonempty — junk cannot weaken it" } ], "flagReadings": [ { "challenge": "QuantitativeVanDerWaerden", "flag": "JUNK? (not triggered): W is sInf on ℕ, which is 0 for an empty set; the strict lower bound by a positive real is false at 0, so the statement itself forces the Ramsey set to be nonempty — junk cannot weaken it", "decision": "cleared", "why": "By van der Waerden's theorem, for every r ≥ 1 and every k there is N > 0 such that every colouring c : ℕ → Fin r has a monochromatic k-term AP with d > 0 and last term < N, which is exactly IsRamsey (Fin r) k N. So the set under sInf is nonempty for every r ≥ 2 the statement quantifies over, and W r k is the true least Ramsey size; the sInf ∅ = 0 branch is never taken. Independently, had it been taken, the claim k^(…) < 0 would be false (a positive base, or 0^0 = 1 at k = 0), so it could only falsify. Lane K: CERTIFIED (run 38019899278).", "evidence": [ "lean/ComparatorChallenges/QuantitativeVanDerWaerden.lean:20: noncomputable def W (r k : ℕ) : ℕ :=", "lean/ComparatorChallenges/QuantitativeVanDerWaerden.lean:21: sInf {N : ℕ | 0 < N ∧ IsRamsey (Fin r) k N}", "lean/ComparatorChallenges/QuantitativeVanDerWaerden.lean:28: (k : ℝ) ^ ((1 / 100000 : ℝ) * k * (Nat.log 2 r : ℝ)) < (W r k : ℝ) := by" ] } ], "readings": [ "first/batch1.json", "flags/flags1.json" ] }, { "family": "161", "title": "Counterexamples to Sidorenko’s conjecture and the forcing conjecture", "headline": "Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness.", "challenges": [ "SidorenkoCounterexample" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density.", "coveredBy": [ "SidorenkoCounterexample" ], "note": "∃ finite simple host with an edge and t(H, host) < t(K₂, host)^66, H the explicit 13+22-vertex incidence graph." }, { "clause": "The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness.", "coveredBy": [], "note": "Not stated." } ], "gap": "The forcing-conjecture sentence.", "scopeNoteOnGap": "lean/docs/161.md:9: \"The separate forcing-conjecture consequence in the paper is outside this statement.\"", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "SidorenkoCounterexample", "flag": "NOTE: connectivity of H is part of the headline but is not asserted in Lean (H is a fixed concrete graph); I checked it on the face table: 35 vertices, 66 edges, connected" } ], "flagReadings": null, "readings": [ "first/batch6.json" ] }, { "family": "162", "title": "Counterexamples to Ryser’s covering conjecture", "headline": "Disproves Ryser's covering conjecture by constructing intersecting $`(q+1)`$-partite, $`(q+1)`$-uniform hypergraphs with covering number $`q+1`$, rather than the predicted bound q, for every sufficiently large prime q. A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture.", "challenges": [ "BalancedRyser", "RyserCovering", "RyserOddExtensions" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Disproves Ryser's covering conjecture by constructing intersecting $`(q+1)`$-partite, $`(q+1)`$-uniform hypergraphs with covering number $`q+1`$, rather than the predicted bound q, for every sufficiently large prime q.", "status": "stated", "coveredBy": [ "BalancedRyser" ], "evidence": [ "lean/ComparatorChallenges/BalancedRyser.lean:33: ∃ H : PartiteHypergraph (q + 1) (q + 1),", "lean/ComparatorChallenges/BalancedRyser.lean:34: H.Intersecting ∧ H.Nonisolated ∧ H.HasCoverNumber (q + 1)", "lean/ComparatorChallenges/BalancedRyser.lean:37: ∃ q₀ : ℕ, ∀ q : ℕ, q₀ ≤ q → q.Prime → BalancedCounterexample q" ], "note": "Edges are functions Fin (q+1) → Fin (q+1) (one vertex per part, hence (q+1)-partite and (q+1)-uniform); HasCoverNumber k = a cover of size k exists and every cover has size ≥ k." }, { "clause": "A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture.", "status": "undeclared", "coveredBy": [], "manuscript": "A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026", "evidence": [ "preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/paper.tex:168: \\begin{corollary}\\label{cor:tree-cover}", "preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/paper.tex:171: put $r=s^n+1$. There is a finite simple complete graph with an", "preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/paper.tex:172: $r$-edge-coloring for which the minimum number of monochromatic trees", "preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/paper.tex:201: Thus Gy\\'arf\\'as's cover conjecture fails at these extension-field ranks.", "lean/docs/162.md:6: - [A counterexample to Ryser's covering conjecture](../../preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf)", "lean/docs/162.md:12: Equal part sizes and numerical thresholds are not asserted.", "lean/ComparatorChallenges/RyserOddExtensions.lean:66: Specification.ExplicitFailureAt (prime ^ degree + 1)) ∧", "lean/ComparatorChallenges/RyserCovering.lean:52: CounterexampleRank (p ^ n + 1) := by" ], "note": "The extension-field hypergraph counterexamples (ranks p^n+1) ARE stated by RyserCovering and RyserOddExtensions; the Gyárfás tree-cover disproof is not — it follows only through Gyárfás's hypergraph–edge-colouring correspondence (the manuscript's Corollary cor:tree-cover, paper.tex:168-199), which is not formalized. Both manuscripts are linked, and the note names only equal part sizes and numerical thresholds as not asserted, never tree covers or Gyárfás." } ], "gap": "Disproof of Gyárfás's monochromatic tree-cover conjecture (an r-edge-coloured complete graph needing r monochromatic trees) is claimed but not stated; only the hypergraph counterexamples it is derived from are stated.", "scopeNoteOnGap": "silent", "note": "Ryser is stated (BalancedRyser, plus RyserCovering/RyserOddExtensions for the extension-field ranks); the Gyárfás monochromatic tree-cover clause is a few-line corollary of the stated hypergraphs via Gyárfás's correspondence, unformalized, from a linked manuscript, and the note is silent on it: NARROWER — UNDECLARED. Agree with the first reader; the gap is small in content (an elementary reduction proved in the paper) but it is a separate disproof the headline names.", "firstSuspicious": "The omitted clause is a one-paragraph corollary of the stated hypergraph counterexamples (Gyárfás's classical equivalence; paper.tex:168-196). A second reader may treat it as a reformulation rather than a gap; it is classed UNDECLARED because the headline names it as a separate disproof, no Comparator statement states it, and the scope note is silent on it.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [ { "challenge": "RyserCovering", "flag": "NARROWER — UNDECLARED (family level): the Gyárfás tree-cover clause is not stated here or elsewhere; lean/docs/162.md does not mention Gyárfás or tree covers." } ], "flagReadings": null, "readings": [ "first/batch3.json", "second/second7.json" ] }, { "family": "165", "title": "The Harary–Hill and Zarankiewicz crossing-number formulas", "headline": "Resolves the Harary–Hill conjecture and Turán's brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs.", "challenges": [ "CompleteCrossing", "BipartiteCrossing" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the Harary–Hill conjecture", "coveredBy": [ "CompleteCrossing" ], "note": "For every order n ≥ 3, ordinaryCrossingNumber (K_n) = ⌊⌊n/2⌋⌊(n−1)/2⌋⌊(n−2)/2⌋⌊(n−3)/2⌋/4⌋ (the outer ℕ division is exact since the product is divisible by 4). The infimum is over the range of crossingCount on admissible drawings; for n ≥ 5 the value is positive, which rules out the sInf ∅ = 0 junk and, ℕ being well-ordered, means the minimum is attained." }, { "clause": "Turán's brickyard problem in the Zarankiewicz formulation", "coveredBy": [ "BipartiteCrossing" ], "note": "For all m, n > 0: some admissible drawing of K_{m,n} has exactly Z(m)Z(n) crossings, and every admissible drawing has ≥ Z(m)Z(n), with Z(r) = ⌊r/2⌋⌊(r−1)/2⌋." }, { "clause": "determining the crossing numbers of every complete and complete bipartite graph", "coveredBy": [ "CompleteCrossing", "BipartiteCrossing" ], "note": "Complete graphs only for n ≥ 3; K_1 and K_2 have fewer than two edges, so their crossing number is trivially 0 = the formula (an omission of trivial cases, written in the scope note as 'for every $n\\ge3$', not a substantive gap). Bipartite: every m, n ≥ 1." }, { "clause": "The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs", "coveredBy": [ "CompleteCrossing", "BipartiteCrossing" ], "note": "Lower bounds hold for every admissible drawing: injective vertices, edges as injective continuous Paths whose interiors avoid vertices, finitely many crossing points, each a locally transversal crossing (an OpenPartialHomeomorph sends the two edge interiors to the two axes), no point on three edge interiors. Crossings are counted as points, which equals the count of crossing edge-pairs because triple points are excluded." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": "A strict reader could call 165 NARROWER — DECLARED because CompleteCrossing assumes 3 ≤ order while the headline says 'every complete … graph' (abstract: 'for every positive integer n'); the omitted K_1, K_2 have no pair of edges, so their crossing number is 0 = the formula in the same Lean model. I judged this an omission of trivial cases, not a gap.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "167", "title": "Planar distinct distances and unit-distance bounds", "headline": "Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but $`o(n)`$ points of any n-point planar set determine at least $`n^{1-\\varepsilon}`$ distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by $`O(n^{4/3-\\delta})`$ for an absolute δ > 0.", "challenges": [ "PinnedDistances", "PlanarUnitDistances" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but $`o(n)`$ points of any n-point planar set determine at least $`n^{1-\\varepsilon}`$ distinct nonzero distances.", "coveredBy": [ "PinnedDistances" ], "note": "Implied by one counting step, not stated verbatim: a pin with < n^{1-ε} distances has ≥ n-1-n^{1-ε/2} partners y whose distance class has ≥ n^{ε/2} points, so (bad pins)/n ≤ F(n,ε/2)/(1-o(1)) → 0, uniformly over n-point sets." }, { "clause": "A complementary theorem bounds the number of unit-distance pairs by $`O(n^{4/3-\\delta})`$ for an absolute δ > 0.", "coveredBy": [ "PlanarUnitDistances" ], "note": "Stated: ∃ C>0, β ∈ [1,4/3), ∀n, u(n) ≤ C n^β." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": "Cosmetic: the scope note contradicts itself — lean/docs/167.md:10: The formalized result shows that large repeated distance fibers are rare in arbitrary finite planar point sets. For each fixed $s>0$, the largest possible fraction of ordered distinct pairs $(x,y)$ whose distance from the pin $x$ occurs at least $n^s$ times tends to zero as the set size $n$ grows. Consequently, for every $\\varepsilon>0$, the fraction of pins determining fewer than $n^{1-\\varepsilon}$ distances tends uniformly to zero. The separate unit-distance power-saving theorem is not included. versus lean/docs/167.md:12: The planar unit-distance problem asks how many pairs at distance one can occur among $n$ points. The formalization proves that there are absolute constants $C>0$ and $1\\le\\beta<4/3$ such that every finite planar point set of size $n$ determines at most $Cn^\\beta$ unordered unit-distance pairs. The same constants work for every $n$, giving a fixed power improvement over the classical exponent $4/3$. and the comparator table row lean/docs/167.md:19: | Power saving for planar unit distances | [PlanarUnitDistances.lean](../ComparatorChallenges/PlanarUnitDistances.lean) | (two per-paper notes concatenated). Not a gap.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "PinnedDistances", "flag": "JUNK? (not triggered): F is a real sSup; the set of fractions lies in [0,1] and is nonempty for n ≥ 2 (F := 0 for n < 2), so it is the true supremum" }, { "challenge": "PlanarUnitDistances", "flag": "JUNK? (not triggered): u(n) is sSup on ℕ, which Mathlib sets to 0 for an unbounded set; here unitPairCount X ≤ |X.sym2| = n(n+1)/2, so the set is bounded and u(n) is the true maximum" } ], "flagReadings": [ { "challenge": "PinnedDistances", "flag": "JUNK? (not triggered): F is a real sSup; the set of fractions lies in [0,1] and is nonempty for n ≥ 2 (F := 0 for n < 2), so it is the true supremum", "decision": "cleared", "why": "F n s (n ≥ 2) is Real.sSup of {pairFraction P s | P.card = n}. richPairs P s is a filter of the off-diagonal of P ×ˢ P, which has n(n−1) elements, and the denominator is n(n−1) > 0, so every pairFraction lies in [0,1]; the plane contains n-point sets for every n, so the set is nonempty. Nonempty and bounded above, so sSup is the true supremum. The junk value (sSup = 0 for an empty or unbounded set) is the weakening direction here, because it would make 'F → 0' trivially true. It cannot be reached. The n < 2 branch is a finite prefix and does not affect a limit at atTop.", "evidence": [ "lean/ComparatorChallenges/PinnedDistances.lean:18: def richPairs (P : Finset Plane) (s : ℝ) : Finset (Plane × Plane) :=", "lean/ComparatorChallenges/PinnedDistances.lean:19: (P ×ˢ P).filter (fun xy => xy.1 ≠ xy.2 ∧", "lean/ComparatorChallenges/PinnedDistances.lean:20: (P.card : ℝ) ^ s ≤ (k P xy.1 xy.2 : ℝ))", "lean/ComparatorChallenges/PinnedDistances.lean:22: def pairFraction (P : Finset Plane) (s : ℝ) : ℝ :=", "lean/ComparatorChallenges/PinnedDistances.lean:23: (richPairs P s).card / ((P.card : ℝ) * ((P.card : ℝ) - 1))", "lean/ComparatorChallenges/PinnedDistances.lean:26: if 2 ≤ n then sSup {a : ℝ | ∃ P : Finset Plane, P.card = n ∧ a = pairFraction P s}" ] }, { "challenge": "PlanarUnitDistances", "flag": "JUNK? (not triggered): u(n) is sSup on ℕ, which Mathlib sets to 0 for an unbounded set; here unitPairCount X ≤ |X.sym2| = n(n+1)/2, so the set is bounded and u(n) is the true maximum", "decision": "cleared", "why": "u n is Nat.sSup, which is 0 for a set not bounded above. That would make 'u n ≤ C n^β' trivially true, so this junk value is the weakening direction. It cannot be reached: unitPairCount X is the card of a filter of X.sym2, so it is ≤ X.sym2.card = n(n+1)/2 (Finset.card_sym2). Every n-point planar set exists, so the set is nonempty and bounded, and u n is the attained maximum. Diagonal elements s(x,x) of sym2 have dist 0 ≠ 1, so they do not inflate the count.", "evidence": [ "lean/ComparatorChallenges/PlanarUnitDistances.lean:10: Sym2.lift ⟨fun x y => dist x y = 1, fun x y => by dsimp; rw [dist_comm]⟩", "lean/ComparatorChallenges/PlanarUnitDistances.lean:14: exact (X.sym2.filter IsUnitPair).card", "lean/ComparatorChallenges/PlanarUnitDistances.lean:17: sSup {m : ℕ | ∃ X : Finset Plane, X.card = n ∧ unitPairCount X = m}", "lean/ComparatorChallenges/PlanarUnitDistances.lean:19: theorem main : ∃ C β : ℝ, 0 < C ∧ 1 ≤ β ∧ β < (4 : ℝ) / 3 ∧", "lean/ComparatorChallenges/PlanarUnitDistances.lean:20: ∀ n : ℕ, (u n : ℝ) ≤ C * (n : ℝ) ^ β := by" ] } ], "readings": [ "first/batch1.json", "flags/flags2.json" ] }, { "family": "168", "title": "Combinatorial invariance of Kazhdan–Lusztig polynomials", "headline": "Resolves the full combinatorial invariance conjecture: isomorphic Bruhat intervals in arbitrary Coxeter systems have identical equal-parameter Kazhdan–Lusztig polynomials. Thus the abstract order of the interval determines the polynomial, even across different Coxeter systems.", "challenges": [ "KLInvariance" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the full combinatorial invariance conjecture: isomorphic Bruhat intervals in arbitrary Coxeter systems have identical equal-parameter Kazhdan–Lusztig polynomials.", "coveredBy": [ "KLInvariance" ], "note": "For arbitrary Coxeter systems cs, cs' (different generator types and universes), u ≤ b, u' ≤ b', and an order isomorphism [u,b] ≃o [u',b']: P_{u,b} = P'_{u',b'}." }, { "clause": "Thus the abstract order of the interval determines the polynomial, even across different Coxeter systems.", "coveredBy": [ "KLInvariance" ], "note": "Same theorem; cs and cs' are independent." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "KLInvariance", "flag": "DEFINITION carries the conclusion? checked, cleared: Classical.epsilon selects SOME family satisfying NormalizedKL. The normalization determines R by the descent recursion and P by the degree bound plus reciprocity (standard induction on length), so the selected family is the KL family. If the normalization were unsatisfiable, the theorem would concern arbitrary polynomials and be unprovable, not trivial." }, { "challenge": "KLInvariance", "flag": "JUNK? checked, none: the finsum would be 0 on an infinite interval, but Bruhat intervals are finite." } ], "flagReadings": [ { "challenge": "KLInvariance", "flag": "DEFINITION carries the conclusion? checked, cleared: Classical.epsilon selects SOME family satisfying NormalizedKL. The normalization determines R by the descent recursion and P by the degree bound plus reciprocity (standard induction on length), so the selected family is the KL family. If the normalization were unsatisfiable, the theorem would concern arbitrary polynomials and be unprovable, not trivial.", "decision": "cleared", "why": "Classical.epsilon picks the KL family, because NormalizedKL has exactly one solution. Existence: the genuine KL pair satisfies every field. R_{xx}=1; R=0 off the order. The left-descent recursion follows from the usual right one via R_{x,y}=R_{x⁻¹,y⁻¹}, and it holds for ALL x, including x ≰ y, by the lifting property. P_{xx}=1; P=0 off the order. deg P_{xy} ≤ (ℓ(y)−ℓ(x)−1)/2 ⇔ 2·deg < d. And q^d P_{xy}(q⁻¹) = Σ_{x≤z≤y} R_{xz}P_{zy}, which is KL 1979 (2.2.c); `reflect d` equals q^d p(q⁻¹) here because natDegree ≤ d. Uniqueness: R is forced by induction on ℓ(y), because every y ≠ 1 has a left descent, and y = 1 is fixed by R_diagonal/R_zero (x ≤ 1 only for x = 1). P is forced by induction on d = ℓ(y)−ℓ(x): reflect d P_{xy} − P_{xy} = Σ_{x d/2 and < d/2 by P_degree), so P_{xy} is determined. BruhatLE is the genuine strong Bruhat order: the reflexive-transitive closure of length-increasing multiplication by a reflection. The definition therefore names the true KL polynomial and carries nothing; the theorem still asserts full combinatorial invariance. Lane K: not run yet.", "evidence": [ "ComparatorChallenges/KLInvariance.lean:14: def BruhatStep (cs : CoxeterSystem M W) (x y : W) : Prop :=", "ComparatorChallenges/KLInvariance.lean:15: cs.length x < cs.length y ∧ ∃ t : W, cs.IsReflection t ∧ y = t * x", "ComparatorChallenges/KLInvariance.lean:18: def BruhatLE (cs : CoxeterSystem M W) : W → W → Prop :=", "ComparatorChallenges/KLInvariance.lean:19: Relation.ReflTransGen (BruhatStep cs)", "ComparatorChallenges/KLInvariance.lean:55: structure NormalizedKL (cs : CoxeterSystem M W) (RP : PolynomialFamilies W) : Prop where", "ComparatorChallenges/KLInvariance.lean:56: R_diagonal : ∀ x, RP.1 x x = 1", "ComparatorChallenges/KLInvariance.lean:57: R_zero : ∀ x y, ¬ BruhatLE cs x y → RP.1 x y = 0", "ComparatorChallenges/KLInvariance.lean:58: R_recursion : ∀ x y i, cs.length (cs.simple i * y) < cs.length y →", "ComparatorChallenges/KLInvariance.lean:59: RP.1 x y =", "ComparatorChallenges/KLInvariance.lean:60: if cs.length (cs.simple i * x) < cs.length x then", "ComparatorChallenges/KLInvariance.lean:61: RP.1 (cs.simple i * x) (cs.simple i * y)", "ComparatorChallenges/KLInvariance.lean:62: else", "ComparatorChallenges/KLInvariance.lean:63: (X - 1) * RP.1 x (cs.simple i * y) +", "ComparatorChallenges/KLInvariance.lean:64: X * RP.1 (cs.simple i * x) (cs.simple i * y)", "ComparatorChallenges/KLInvariance.lean:65: P_diagonal : ∀ x, RP.2 x x = 1", "ComparatorChallenges/KLInvariance.lean:66: P_zero : ∀ x y, ¬ BruhatLE cs x y → RP.2 x y = 0", "ComparatorChallenges/KLInvariance.lean:67: P_degree : ∀ x y, BruhatLE cs x y → x ≠ y →", "ComparatorChallenges/KLInvariance.lean:68: 2 * (RP.2 x y).natDegree < rankDifference cs x y", "ComparatorChallenges/KLInvariance.lean:69: reciprocity : ∀ x y, BruhatLE cs x y →", "ComparatorChallenges/KLInvariance.lean:70: reflect (rankDifference cs x y) (RP.2 x y) =", "ComparatorChallenges/KLInvariance.lean:71: ∑ᶠ z : Interval cs x y, RP.1 x z.val * RP.2 z.val y", "ComparatorChallenges/KLInvariance.lean:74: noncomputable def klFamilies (cs : CoxeterSystem M W) : PolynomialFamilies W :=", "ComparatorChallenges/KLInvariance.lean:75: Classical.epsilon (NormalizedKL cs)", "ComparatorChallenges/KLInvariance.lean:76: ", "ComparatorChallenges/KLInvariance.lean:77: noncomputable def klPolynomial (cs : CoxeterSystem M W) (x y : W) : ℤ[X] :=", "ComparatorChallenges/KLInvariance.lean:78: (klFamilies cs).2 x y" ] }, { "challenge": "KLInvariance", "flag": "JUNK? checked, none: the finsum would be 0 on an infinite interval, but Bruhat intervals are finite.", "decision": "cleared", "why": "∑ᶠ over `Interval cs x y` would be 0 only if the support were infinite. Bruhat intervals in a Coxeter group are finite: by the subword property every z ≤ y is a subword of one fixed reduced word of y, so there are at most 2^ℓ(y) of them. The finsum is the genuine finite sum, and the reciprocity field is the true identity. (Had it been junk, NormalizedKL would become unsatisfiable or non-standard and the theorem would be about an arbitrary epsilon-chosen family, so it would be unprovable, not trivial.)", "evidence": [ "ComparatorChallenges/KLInvariance.lean:22: def Interval (cs : CoxeterSystem M W) (u b : W) :=", "ComparatorChallenges/KLInvariance.lean:23: {x : W // BruhatLE cs u x ∧ BruhatLE cs x b}", "ComparatorChallenges/KLInvariance.lean:69: reciprocity : ∀ x y, BruhatLE cs x y →", "ComparatorChallenges/KLInvariance.lean:70: reflect (rankDifference cs x y) (RP.2 x y) =", "ComparatorChallenges/KLInvariance.lean:71: ∑ᶠ z : Interval cs x y, RP.1 x z.val * RP.2 z.val y" ] } ], "readings": [ "first/batch5.json", "flags/flags3.json" ] }, { "family": "169", "title": "Shareshian–Wachs elementary positivity", "headline": "Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in $`\\mathbb N[q]`$. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity.", "challenges": [ "ElementaryPositivity" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in $`\\mathbb N[q]`$", "coveredBy": [ "ElementaryPositivity" ], "note": "For every n and every Hessenberg function h (monotone, i ≤ h i) the definition hole's type G.PermutationWitness requires, for every number r of variables, X_G(x_1..x_r; q) = Σ_{σ nondescent} q^{inv_G(σ)} · e_{θ(σ)} in MvPolynomial (Fin r) ℕ[q]; grouping by θ(σ) gives ℕ[q] coefficients in the e-basis. X_G is the Shareshian–Wachs sum over proper colourings of q^{asc(f)} x^f. Holding for every finite r is equivalent to the symmetric-function identity." }, { "clause": "The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity", "coveredBy": [ "ElementaryPositivity" ], "note": "The counted objects (permutations whose consecutive descents are graph edges) and their q-weights (graph inversions) are explicit in the statement, but the partition label θ : nondescent permutations → partitions of n is EXISTENTIAL (a field of the witness), so the statement fixes the coefficient of e_λ only as the q-count of the fibre θ⁻¹(λ) for some θ." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": [ { "challenge": "ElementaryPositivity", "strict": "SAME BODIES" } ], "flags": [ { "challenge": "ElementaryPositivity", "flag": "DEFINITION HOLE: definition_names = [OAI.elementaryPositivityWitness]; the challenge has no theorem_names. Here the hole's TYPE (G.PermutationWitness, a structure whose 'expansion' field is the Prop equation) carries the entire claim, so any inhabitant proves it and an unchecked body cannot weaken the statement; only the axiom audit matters for this hole." }, { "challenge": "ElementaryPositivity", "flag": "QUANTIFIER?: θ is existential inside the witness, so 'explicitly described permutations' is stated as 'some partition labelling of the nondescent permutations' — the explicit labelling of the paper is not pinned by the statement (the positivity claim is unaffected)." } ], "flagReadings": [ { "challenge": "ElementaryPositivity", "flag": "QUANTIFIER?: θ is existential inside the witness, so 'explicitly described permutations' is stated as 'some partition labelling of the nondescent permutations' — the explicit labelling of the paper is not pinned by the statement (the positivity claim is unaffected).", "decision": "cleared", "why": "The labelling θ is existential, but this does not drop any truth condition of the headline. The statement fixes the counted objects explicitly: permutations whose consecutive descents are G-edges (Nondescent). It fixes their q-weights (graphInversions) and requires that the fibres of θ partition them into the e_λ coefficients for every number r of variables, which pins the symmetric-function identity. So the ℕ[q] e-positivity (sentence 1) is fully stated, and so is the content of sentence 2: each coefficient is a q-count of a set of explicitly defined permutations with an explicit statistic, and the sets partition the nondescent permutations. That is stronger than bare positivity, since it fixes Σ_λ c_λ(q). What is left open is the paper's particular rule for θ, which is a property of the proof, not a mathematical claim. θ ranges over a finite set, and the identity is decidable (it suffices at r = n), so a θ exists iff a computable one does (the first in any enumeration). 'Explicitly described' therefore adds no truth condition the Lean misses. The witness is a def, so the solution supplies an actual θ; lane K: CERTIFIED, strict run SAME BODIES.", "evidence": [ "ComparatorChallenges/ElementaryPositivity.lean:22-23: def Nondescent (σ : Equiv.Perm (Fin n)) : Prop := ⏎ ∀ i j : Fin n, i.val + 1 = j.val → σ j < σ i → G.Edge (σ i) (σ j)", "ComparatorChallenges/ElementaryPositivity.lean:29-31: def graphInversions (σ : Equiv.Perm (Fin n)) : ℕ := ⏎ (Finset.univ.filter fun ij : Fin n × Fin n => ⏎ ij.1 < ij.2 ∧ σ ij.2 < σ ij.1 ∧ G.Edge (σ ij.1) (σ ij.2)).card", "ComparatorChallenges/ElementaryPositivity.lean:48-53: structure PermutationWitness where ⏎ theta : {σ : Equiv.Perm (Fin n) // G.Nondescent σ} → Nat.Partition n ⏎ expansion : ∀ r : ℕ, G.chromatic r = ⏎ ∑ σ : {σ : Equiv.Perm (Fin n) // G.Nondescent σ}, ⏎ MvPolynomial.C (Polynomial.X ^ G.graphInversions σ.val) * ⏎ MvPolynomial.esymmPart (Fin r) (Polynomial ℕ) (theta σ)", "ComparatorChallenges/ElementaryPositivity.lean:58-60: def elementaryPositivityWitness (n : ℕ) ⏎ (G : ElementaryPositivity.NaturalUnitIntervalGraph n) : ⏎ G.PermutationWitness := by" ] } ], "readings": [ "first/batch2.json", "flags/flags4.json" ] }, { "family": "170", "title": "Sharp logarithmic exponents for off-diagonal Ramsey numbers", "headline": "For every fixed integer s ≥ 5, proves $`r(s,t)=t^{s-1}/(\\log t)^{s-2+o(1)}`$ as $`t\\to\\infty`$, determining the logarithmic exponent and matching the classical upper bound at that scale. Here $`r(s,t)`$ is the least number of vertices forcing an s-clique or a t-vertex independent set.", "challenges": [ "RamseyFive", "SharpLogRamsey" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "For every fixed integer s ≥ 5, proves $`r(s,t)=t^{s-1}/(\\log t)^{s-2+o(1)}`$ as $`t\\to\\infty`$", "coveredBy": [ "RamseyFive", "SharpLogRamsey" ], "note": "s = 5 in RamseyFive and every s >= 6 in SharpLogRamsey. Bounds: for each eps > 0, eventually t^(s-1)/(log t)^(s-2+eps) <= r(s,t) <= C t^(s-1)/(log t)^(s-2), with C independent of eps; plus the limit (( s-1) log t - log r)/log log t -> s - 2." }, { "clause": "determining the logarithmic exponent and matching the classical upper bound at that scale", "coveredBy": [ "RamseyFive", "SharpLogRamsey" ], "note": "Limit statements SharpExponent and MainLimit; the upper bound has the classical shape C t^(s-1)/(log t)^(s-2)." }, { "clause": "Here $`r(s,t)`$ is the least number of vertices forcing an s-clique or a t-vertex independent set", "coveredBy": [ "RamseyFive", "SharpLogRamsey" ], "note": "ramsey s t = sInf {N | every simple graph on Fin N has an s-clique or an independent t-set}." } ], "gap": null, "scopeNoteOnGap": "no gap", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "172", "title": "Classification of finite Euclidean Ramsey configurations", "headline": "Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space. The characterization is an algebraic condition over the coordinate field. It also disproves the Leader–Russell–Walters conjecture that every such configuration is a subset of a finite transitive set.", "challenges": [ "EuclideanRamsey", "GrahamSpherical", "EuclideanRamseyCircle", "EuclideanRamseyNine", "EuclideanRamseyQuadratic", "EuclideanRamseySpherical", "EuclideanRamseyTransitive" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space.", "status": "stated", "coveredBy": [ "EuclideanRamsey" ], "evidence": [ "lean/ComparatorChallenges/EuclideanRamsey.lean:14: def Ramsey {s d : ℕ} (a : Fin s → Space d) : Prop :=", "lean/ComparatorChallenges/EuclideanRamsey.lean:15: ∀ r : ℕ, 2 ≤ r → ∃ D : ℕ, 1 ≤ D ∧", "lean/ComparatorChallenges/EuclideanRamsey.lean:16: ∀ c : Space D → Fin r, ∃ b : Fin s → Space D,", "lean/ComparatorChallenges/EuclideanRamsey.lean:17: Congruent a b ∧ ∃ k : Fin r, ∀ i, c (b i) = k", "lean/ComparatorChallenges/EuclideanRamsey.lean:42: theorem classification {s d : ℕ} (a : Fin s → Space d)", "lean/ComparatorChallenges/EuclideanRamsey.lean:43: (hs : 2 ≤ s) (hd : 1 ≤ d) (ha : Function.Injective a)", "lean/ComparatorChallenges/EuclideanRamsey.lean:44: (hspan : affineSpan ℝ (Set.range a) = ⊤) :", "lean/ComparatorChallenges/EuclideanRamsey.lean:45: Ramsey a ↔ FieldCriterion a := by", "encoding: stated for injective configurations of at least 2 points with full affine span; Ramsey depends only on pairwise distances (Congruent, EuclideanRamsey.lean:11-12), so any finite configuration is congruent to a full-span one and singletons are trivially Ramsey", "lean/OAI/Combinatorics/EuclideanRamsey/Main.lean:78: theorem full_affine_representative {s D : ℕ} (a : Fin s → Space D)", "lean/OAI/Combinatorics/EuclideanRamsey/Congruence.lean:13: theorem singleton_ramsey {d : ℕ} (a : Fin 1 → Space d) : Ramsey a := by" ] }, { "clause": "The characterization is an algebraic condition over the coordinate field.", "status": "stated", "coveredBy": [ "EuclideanRamsey" ], "evidence": [ "lean/ComparatorChallenges/EuclideanRamsey.lean:19: def coordinateField {s d : ℕ} (a : Fin s → Space d) : IntermediateField ℚ ℝ :=", "lean/ComparatorChallenges/EuclideanRamsey.lean:20: IntermediateField.adjoin ℚ (Set.range (fun ij : Fin s × Fin d => a ij.1 ij.2))", "lean/ComparatorChallenges/EuclideanRamsey.lean:35: def FieldCriterion {s d : ℕ} (a : Fin s → Space d) : Prop :=", "lean/ComparatorChallenges/EuclideanRamsey.lean:36: ∃ P : Matrix (Option (Fin d)) (Option (Fin d)) (TensorRing a),", "lean/ComparatorChallenges/EuclideanRamsey.lean:37: (∀ i : Fin s, ∑ α, ∑ β,", "lean/ComparatorChallenges/EuclideanRamsey.lean:38: ((augmented a i α) ⊗ₜ[ℚ] (1 : Coeff a)) * P α β *", "lean/ComparatorChallenges/EuclideanRamsey.lean:39: ((1 : Coeff a) ⊗ₜ[ℚ] (augmented a i β)) = 0) ∧", "lean/ComparatorChallenges/EuclideanRamsey.lean:40: (∀ α β : Fin d, multiply a (P (some α) (some β)) = if α = β then 1 else 0)" ] }, { "clause": "It also disproves the Leader–Russell–Walters conjecture that every such configuration is a subset of a finite transitive set.", "status": "undeclared", "coveredBy": [ "EuclideanRamseyCircle" ], "manuscript": "A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026", "evidence": [ "preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/build/sections/07-consequences.tex:130: is Ramsey and is not subtransitive.", "preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/build/sections/07-consequences.tex:135: Leader, Russell and Walters proved that $K_a$ does not embed in any", "preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/build/sections/07-consequences.tex:136: finite transitive Euclidean set \\cite[Corollary~2]{lrw-kites}.", "lean/ComparatorChallenges/EuclideanRamseyCircle.lean:21: theorem at_most_five_circle_points_ramsey {s : ℕ} (a : Fin s → Space 2)", "lean/ComparatorChallenges/EuclideanRamseyCircle.lean:22: (hs : 0 < s) (hs5 : s ≤ 5) (ha : Function.Injective a)", "lean/ComparatorChallenges/EuclideanRamseyCircle.lean:23: (hsphere : ∃ c : Space 2, ∃ r : ℝ, ∀ i, dist (a i) c = r) : Ramsey a := by", "lean/docs/172.md:5: - [A classification of finite Euclidean Ramsey configurations](../../preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf)", "lean/docs/172.md:9: It also proves that every nonempty subset of a finite transitive configuration and every nonempty set of at most five points on a circle is Ramsey.", "lean/docs/172.md: no mention of Leader–Russell–Walters, subtransitivity of the kites, or the disproof" ] } ], "gap": "The existence of a Ramsey configuration that is not subtransitive — the actual counterexample to the LRW conjecture. Its non-subtransitivity half is a cited prior theorem in the paper, not proved there and not in Lean.", "scopeNoteOnGap": "silent", "note": "The classification is stated (full-affine-span normalization named as an encoding). The LRW disproof needs a Ramsey set that is NOT subtransitive: Lean states only the Ramsey half (≤5 concyclic points), and the non-subtransitivity of the kites K_a is Leader–Russell–Walters 2011 (J. Comb. 2(3), Cor. 2), cited by the paper and not formalized; the scope note is silent, so UNDECLARED. Flag CREDIT OUTSIDE THE CLOSURE: 172.md:9 credits the formalization with the singleton and affine-span reductions, whose theorems (singleton_ramsey, full_affine_representative) are in OAI/ but in no challenge. Agree with the first reader.", "firstSuspicious": "Judgment call for a second reader: the unstated half is a published theorem the paper cites rather than proves; under a rule that tolerates named external inputs this family would read MATCHES.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch1.json", "second/second1.json" ] }, { "family": "173", "title": "Seymour’s second-neighborhood conjecture", "headline": "Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely directed edge pairs.", "challenges": [ "SeymourSecondNeighborhood" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one.", "coveredBy": [ "SeymourSecondNeighborhood" ], "note": "second neighbours exclude v and out-neighbours, so they are exactly the vertices at distance two" }, { "clause": "Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely directed edge pairs.", "coveredBy": [ "SeymourSecondNeighborhood" ], "note": "arbitrary relation r on a Fintype with loopless and asymmetric only" } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch8.json" ] }, { "family": "174", "title": "Deterministic construction of strong thin spanning trees", "headline": "Resolves the strong thin-tree conjecture constructively. Every finite loopless k-edge-connected multigraph on at least two vertices has a spanning tree containing at most a universal $`C/k`$ fraction of the edges of every cut. Such a tree can be found deterministically in polynomial time, even with binary-encoded parallel-edge multiplicities.", "challenges": [ "StrongThinTree", "AlgorithmicThinTrees" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the strong thin-tree conjecture constructively. Every finite loopless k-edge-connected multigraph on at least two vertices has a spanning tree containing at most a universal $`C/k`$ fraction of the edges of every cut.", "coveredBy": [ "StrongThinTree" ], "note": "∃ C > 0 ∀ k ≥ 1 ∀ n ≥ 2, every loopless k-edge-connected multigraph has a spanning tree T with |T ∩ δ(S)| ≤ (C/k)|δ(S)| for every nontrivial S." }, { "clause": "Such a tree can be found deterministically in polynomial time, even with binary-encoded parallel-edge multiplicities.", "coveredBy": [ "AlgorithmicThinTrees" ], "note": "One deterministic multi-stack machine halting within a·(len+1)^degree steps on every valid explicit or binary-multiplicity input, with output a C-thin spanning tree." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch3.json" ] }, { "family": "175", "title": "Talagrand’s expectation thresholds, discrete convexity, and graph decompositions", "headline": "Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli–He–Park–Talagrand graph-decomposition conjecture: every graph's edges split into a universally bounded number of fixed pieces, each with containment threshold at most a universal constant times the original graph's integral expectation threshold. The pieces' embeddings need not agree on shared vertices.", "challenges": [ "TalagrandExpectationThreshold", "TalagrandDiscreteConvexity" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves that integral and fractional expectation thresholds differ by at most a universal factor", "status": "stated", "coveredBy": [ "TalagrandExpectationThreshold" ], "evidence": [ "lean/ComparatorChallenges/TalagrandExpectationThreshold.lean:52: theorem talagrand_expectation_threshold_equivalence [Nonempty α]", "lean/ComparatorChallenges/TalagrandExpectationThreshold.lean:55: qf F ≤ ((25 : ℝ) * (512 : ℝ) ^ 4) * q F := by" ], "note": "q_f ≤ 25·512⁴·q for every nonempty proper increasing family on a finite nonempty ground set. The reverse q ≤ q_f is immediate from the definitions (an integral cover is a fractional cover of the same cost), not a theorem. Both sups are genuine: p = 0 is feasible, since ∅ ∉ F." }, { "clause": "and resolves Talagrand's discrete-convexity conjecture.", "status": "stated", "coveredBy": [ "TalagrandDiscreteConvexity" ], "evidence": [ "lean/ComparatorChallenges/TalagrandDiscreteConvexity.lean:49: (hD : 1 - 1 / (2 : ℝ) ^ 75 ≤ familyMeasure p D) :", "lean/ComparatorChallenges/TalagrandDiscreteConvexity.lean:50: Small p (exceptional ((2 : ℕ) ^ 75) D) := by" ], "note": "With k = 2^75, every 0 < p < 1, every family (no monotonicity), repeats allowed in the union." }, { "clause": "An application proves the Ascoli–He–Park–Talagrand graph-decomposition conjecture: every graph's edges split into a universally bounded number of fixed pieces, each with containment threshold at most a universal constant times the original graph's integral expectation threshold.", "status": "unlinked", "coveredBy": [], "manuscript": "Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026", "evidence": [ "lean/docs/175.md:5: - [Integral and fractional expectation thresholds are equivalent](../../preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026/paper.pdf)", "lean/docs/175.md:6: - [Talagrand's discrete-convexity conjecture](../../preprints/Talagrands-discrete-convexity-conjecture-September-23-2026/paper.pdf)", "CONTENTS.md:4090: We prove the graph-decomposition conjecture of Ascoli, He, Park, and Talagrand." ], "note": "No challenge mentions graphs, containment thresholds or decompositions." }, { "clause": "The pieces' embeddings need not agree on shared vertices.", "status": "unlinked", "coveredBy": [], "manuscript": "Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026", "evidence": [ "CONTENTS.md:4090: the separate embeddings of the pieces need not agree on shared vertices." ], "note": "A qualifier of the unlinked graph-decomposition clause." } ], "gap": "The Ascoli–He–Park–Talagrand graph-decomposition theorem (October-5 manuscript, not linked by the scope note).", "scopeNoteOnGap": "silent", "note": "Both September-23 results are stated. The graph-decomposition application comes from the October-5 manuscript, which the note does not link. The first reader called it UNDECLARED before amendment 3; the gap they found is the same.", "firstSuspicious": "The note links only the two September-23 manuscripts and never mentions graph decompositions.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "TalagrandExpectationThreshold", "flag": "JUNK? (not triggered): q and q_f are real sSups of subsets of [0,1]; both contain p = 0 for a nonempty proper increasing F (∅ ∉ F; singletons cost 0^1 = 0), so the sups are genuine" } ], "flagReadings": null, "readings": [ "first/batch1.json", "second/second2.json" ] }, { "family": "176", "title": "The second Kahn–Kalai conjecture with an edge-count bound", "headline": "Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in $`G(n,p)`$ is at most $`C p_{\\mathrm E}(n,H)(1+\\log_2 h)`$, with universal C. Here $`p_{\\mathrm E}`$ is the least density at which every subgraph of H has expected copy count at least 1/2.", "challenges": [ "SecondKahnKalai" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in $`G(n,p)`$ is at most $`C p_{\\mathrm E}(n,H)(1+\\log_2 h)`$, with universal C", "coveredBy": [ "SecondKahnKalai" ], "note": "For every n ≥ 2, every finite V with |V| ≤ n and every H on V with ≥ 1 edge: p_c ≤ min(1, 2048·e^50·p_E·(1 + log₂ h)) and p_c ≤ 6144·e^50·p_E·log₂ n, where p_c = inf{p ∈ [0,1] : P(G(n,p) ⊇ copy of H) ≥ 1/2}. The constant is explicit and universal." }, { "clause": "Here $`p_{\\mathrm E}`$ is the least density at which every subgraph of H has expected copy count at least 1/2", "coveredBy": [ "SecondKahnKalai" ], "note": "expectationThreshold = inf{p ∈ [0,1] : ∀ F : H.Subgraph, E[copyCount(G(n,p), F)] ≥ 1/2}." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "177", "title": "Bounded-degree coboundary expanders", "headline": "Constructs arbitrarily large finite d-dimensional simplicial complexes, for every d ≥ 3, with uniformly bounded vertex degrees and uniform 𝔽2 coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of such expanders in every positive dimension.", "challenges": [ "CoboundaryExpanders" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs arbitrarily large finite d-dimensional simplicial complexes, for every d ≥ 3, with uniformly bounded vertex degrees and uniform 𝔽2 coboundary expansion in every degree below d.", "coveredBy": [ "CoboundaryExpanders" ], "note": "for each d>=3: a sequence of pure connected d-complexes with vertex counts → ∞, a uniform bound D on top faces per vertex, and one ε>0 for every degree i 1 exactly when $`n\\in\\{2,3,4,5,7,11,13\\}`$.", "challenges": [ "CirculantHadamard", "EvenBarker" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that real circulant Hadamard matrices exist exactly in orders 1 and 4, resolving the circulant Hadamard conjecture.", "coveredBy": [ "CirculantHadamard" ], "note": "Stated as an iff for every n > 0." }, { "clause": "Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most 1 exist at lengths n > 1 exactly when $`n\\in\\{2,3,4,5,7,11,13\\}`$.", "coveredBy": [ "EvenBarker" ], "note": "Only the even-length direction is stated (an even-length Barker sequence has length 2 or 4). The odd-length classification (classical, Turyn–Storer) is declared outside; existence at the seven listed lengths (classical explicit sequences) is not stated and not mentioned." } ], "gap": "Odd Barker lengths and existence at 2,3,4,5,7,11,13.", "scopeNoteOnGap": "The paper's classification of odd Barker lengths is outside this selected additional statement.", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "EvenBarker", "flag": "NARROWER (declared): even lengths only, one direction" } ], "flagReadings": null, "readings": [ "first/batch1.json" ] }, { "family": "180", "title": "Barnette’s Hamiltonian-cycle conjecture", "headline": "Proves that every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture. Equivalently, every three-edge path in a finite simple cubic 3-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle.", "challenges": [ "BarnetteHamiltonian" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Proves that every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture.", "status": "stated", "coveredBy": [ "BarnetteHamiltonian" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/BarnetteHamiltonian.lean:45: G.IsRegularOfDegree 3 → G.IsBipartite → Planar G →", "lean/ComparatorChallenges/BarnetteHamiltonian.lean:46: ThreeVertexConnected G → HasHamiltonianCycle G" ] }, { "clause": "Equivalently, every three-edge path in a finite simple cubic 3-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle.", "status": "undeclared", "coveredBy": [], "manuscript": "Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026", "evidence": [ "preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/build/paper.tex:111: through their published equivalence \\cite[Theorem~4.16]{GSW2023}:", "preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/build/paper.tex:812: \\begin{corollary}[Prescribed paths in Pfaffian graphs]\\label{cor:pfaffian-paths}", "preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/build/paper.tex:821: Gorsky, Steiner, and Wiederrecht \\cite[Theorem~4.16]{GSW2023} proved", "lean/docs/180.md:9: The formalization proves this statement for every such graph. The selected result is the existence of a cycle visiting every vertex exactly once." ] } ], "gap": "the Pfaffian three-edge-path formulation", "scopeNoteOnGap": "silent — docs/180.md:9 says only \"The formalization proves this statement for every such graph. The selected result is the existence of a cycle visiting every vertex exactly once.\"; it does not mention Pfaffian graphs or prescribed paths", "note": "Barnette is stated (topological plane embedding with simple arcs; 3-connected = at least 4 vertices and connected after deleting any 2). The Pfaffian P4-Hamiltonicity clause follows only through Gorsky–Steiner–Wiederrecht 2023, Theorem 4.16 (not formalized), so it is not stated; it is the linked manuscript's Corollary cor:pfaffian-paths, and 'The selected result is …' describes the formalized part without naming the corollary as left out. The gap is a published equivalence, not new mathematics. Agrees with the first reader.", "firstSuspicious": "Borderline: the headline calls the Pfaffian statement 'Equivalently', and the scope note's 'The selected result is the existence of a cycle visiting every vertex exactly once' may be read as declaring the selection. Under the strict rule (the scope note must say what is left out) it is UNDECLARED; a second reader should adjudicate. Mathematically the gap is the external GSW2023 equivalence, not new content.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch8.json", "second/second3.json" ] }, { "family": "181", "title": "The Erdős–Gallai cycle-decomposition conjecture", "headline": "Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most $`Cn`$ simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle-decomposition conjecture, bounding the number of pieces linearly even for dense graphs.", "challenges": [ "CycleDecomposition" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most $`Cn`$ simple cycles and single edges, for an absolute constant C.", "coveredBy": [ "CycleDecomposition" ], "note": "∃ C>0 ∀ n ∀ G on Fin n — correct quantifier order" }, { "clause": "This resolves the Erdős–Gallai cycle-decomposition conjecture, bounding the number of pieces linearly even for dense graphs.", "coveredBy": [ "CycleDecomposition" ], "note": "restatement" } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch8.json" ] }, { "family": "182", "title": "Power savings for intersective polynomial differences and prime arguments", "headline": "For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of $`\\{1,\\ldots,N\\}`$ avoiding nonzero values $`h(1),h(2),\\ldots`$ as differences has size $`O_h(N^{1-c_k})`$, with $`c_k\\gt 0`$ depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when h has a unit root modulo every modulus, with exponent allowed to depend on h.", "challenges": [ "SquareDifference" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of $`\\{1,\\ldots,N\\}`$ avoiding nonzero values $`h(1),h(2),\\ldots`$ as differences has size $`O_h(N^{1-c_k})`$, with $`c_k\\gt 0`$ depending only on degree.", "status": "unlinked", "coveredBy": [], "manuscript": "A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026", "evidence": [ "CONTENTS.md:4220: An integer polynomial is intersective if it has a root modulo every positive integer. For each degree k ≥ 2, we prove that there is an exponent $`c_k\\gt 0`$", "lean/docs/182.md:5: - [A power saving for square-difference-free sets](../../preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf)", "lean/ComparatorChallenges/SquareDifference.lean:8: ∀ a ∈ A, ∀ b ∈ A, ∀ m : ℕ, 1 ≤ m → a - b ≠ (m : ℤ) ^ 2" ], "why": "Only the instance h(x)=x² is stated (absolute c and C, every N ≥ 1); it is implied by the clause (x² is intersective, degree 2, positive leading coefficient), not conversely. The general intersective result is the October 5 manuscript's; the linked square-difference paper does not claim it." }, { "clause": "Here intersective means having a root modulo every modulus.", "status": "unlinked", "coveredBy": [], "manuscript": "A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026", "evidence": [ "CONTENTS.md:4220: An integer polynomial is intersective if it has a root modulo every positive integer." ], "why": "Definitional gloss on the clause above; no intersectivity notion occurs in the statement." }, { "clause": "For prime arguments, a power saving also holds when h has a unit root modulo every modulus, with exponent allowed to depend on h.", "status": "unlinked", "coveredBy": [], "manuscript": "A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026", "evidence": [ "CONTENTS.md:4229: We prove that any set $`A\\subseteq\\{1,\\ldots,N\\}`$ whose differences avoid all nonzero values $`h(p)`$ at primes satisfies $`|A|\\le C_hN^{1-c_h}`$", "lean/docs/182.md:5: - [A power saving for square-difference-free sets](../../preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf)" ] } ], "gap": "Every intersective h of every degree k ≥ 2 (with c_k depending only on k) and the prime-argument version; only the square case (Green–Sawhney's fixed-power question) is formalized.", "scopeNoteOnGap": "silent", "note": "The headline speaks only of the two October 5 generalizations, both unlinked; the linked paper's square-difference theorem is stated exactly and is the h(x)=x² instance of the first clause, so the statement is implied by the headline: NARROWER — UNLINKED, not DIFFERENT. I differ from the first reader (UNDECLARED) only because UNLINKED did not exist for them; their evidence is the same.", "firstSuspicious": "The note links only the square-difference manuscript (2 of the family's 3 manuscripts unlinked).", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "SquareDifference", "flag": "NARROWER (undeclared): the single polynomial x²" } ], "flagReadings": null, "readings": [ "first/batch1.json", "second/second4.json" ] }, { "family": "183", "title": "Power savings for planar halving lines and k-sets", "headline": "Improves the planar halving-line bound to $`O(n^{4/3-\\varepsilon})`$ for sets with no three collinear and an absolute ε > 0. More generally, an n-point set with no three collinear has $`O(n(k+1)^{1/3-\\varepsilon_0})`$ strictly separable k-subsets for $`1\\le k\\le n/2`$, with an absolute $`\\varepsilon_0\\gt 0`$. The constants and positive exponents are nonquantitative.", "challenges": [ "HalvingLines" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Improves the planar halving-line bound to $`O(n^{4/3-\\varepsilon})`$ for sets with no three collinear and an absolute ε > 0.", "status": "stated", "coveredBy": [ "HalvingLines" ], "reading": "first conjunct of power_bounds: ∃ ε > 0, C, n₀ ∀ even n ≥ n₀ ∀ injective P with no three collinear, #halving pairs ≤ C·n^{4/3−ε} (n₀ is absorbed by O since the count is ≤ n²).", "evidence": [ "lean/ComparatorChallenges/HalvingLines.lean:46: (∃ ε : ℝ, 0 < ε ∧ ∃ C : ℝ, ∃ n₀ : ℕ,", "lean/ComparatorChallenges/HalvingLines.lean:48: GeneralPosition P → (halvingCount P : ℝ) ≤ C * (n : ℝ) ^ ((4 : ℝ) / 3 - ε)) ∧" ] }, { "clause": "More generally, an n-point set with no three collinear has $`O(n(k+1)^{1/3-\\varepsilon_0})`$ strictly separable k-subsets for $`1\\le k\\le n/2`$, with an absolute $`\\varepsilon_0\\gt 0`$.", "status": "undeclared", "coveredBy": [], "manuscript": "A-power-saving-for-planar-halving-lines-September-25-2026", "classicalTheorem": "Matoušek's shallow-cutting lemma with the Agarwal–Aronov–Chan–Sharir reduction (Lemma 3.2), plus the manuscript's perturbation from Generic to no-three-collinear — not formalized; the stated k-uniform C·n^{4/3−ε} switch-count bound is weaker than O(n(k+1)^{1/3−ε₀}) for every k = o(n).", "evidence": [ "lean/ComparatorChallenges/HalvingLines.lean:50: ∀ n : ℕ, 1 ≤ n → ∀ P : Fin n → Point, Generic P →", "lean/ComparatorChallenges/HalvingLines.lean:51: ∀ k : ℕ, k ≤ n → (switchCount P k : ℝ) ≤ C * (n : ℝ) ^ ((4 : ℝ) / 3 - ε)) :=", "preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/introduction.tex:96: \\begin{corollary}[Planar $k$-sets and straight-line levels]", "preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/introduction.tex:103: C_0 n(k+1)^{1/3-\\varepsilon_0}.", "preprints/A-power-saving-for-planar-halving-lines-September-25-2026/build/sections/recurrence.tex:121: Apply Matou\\v{s}ek's shallow-cutting lemma in the form stated by", "lean/docs/183.md:5: - [A power saving for planar halving lines](../../preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf)", "lean/docs/183.md:9: It also proves a bound of the same form for all level-switch counts under the additional generic-position assumptions in the statement." ] }, { "clause": "The constants and positive exponents are nonquantitative.", "status": "stated", "coveredBy": [ "HalvingLines" ], "reading": "ε and C are existential in both conjuncts.", "evidence": [ "lean/ComparatorChallenges/HalvingLines.lean:49: (∃ ε : ℝ, 0 < ε ∧ ∃ C : ℝ," ] } ], "gap": "The k-set clause — O(n(k+1)^{1/3−ε₀}) strictly separable k-subsets for 1 ≤ k ≤ n/2 under no-three-collinear — is not stated. The only k-dependent statement is a k-uniform C·n^{4/3−ε} bound on level-switch counts, and it needs additional genericity hypotheses.", "scopeNoteOnGap": "silent — docs/183.md:9 says 'It also proves a bound of the same form for all level-switch counts under the additional generic-position assumptions in the statement.', which describes the weaker k-uniform bound accurately but never says that the headline's O(n(k+1)^{1/3−ε₀}) k-set bound is left out.", "note": "The halving-line power saving is stated. The k-set bound O(n(k+1)^{1/3−ε₀}) is the linked manuscript's Corollary cor:k-sensitive; the Lean states only a k-uniform level-switch bound under extra genericity, from which the k-set bound follows only through the shallow-cutting reduction; docs/183.md describes the switch-count bound and never names the k-set bound as left out. Agrees with the first reader (who noted the k-set sentence is absent from the abstract — it is in the manuscript body).", "firstSuspicious": "The k-set sentence of the headline is also absent from the manuscript abstract (which claims only halving pairs), so it may be a CONTENTS.md overstatement rather than a formalization omission; either way no challenge states it.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json", "second/second5.json" ] }, { "family": "184", "title": "Correspondence coloring with a fixed forbidden subgraph", "headline": "Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need $`O_F(\\Delta/\\log\\Delta)`$ colors when their maximum degree Δ is sufficiently large. Also proves the Ajtai–Erdős–Komlós–Szemerédi independence conjecture: for fixed r ≥ 4, every n-vertex Kr-free graph of average degree d ≥ 2 has an independent set of size $`\\Omega_r(n\\log d/d)`$.", "challenges": [ "CliqueFreeLog" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need $`O_F(\\Delta/\\log\\Delta)`$ colors when their maximum degree Δ is sufficiently large.", "status": "unlinked", "coveredBy": [], "manuscript": "Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026", "evidence": [ "preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026/build/main.tex:9-15: For every fixed integer $r\\ge4$, we prove that every $K_r$-free graph ⏎ of sufficiently large maximum degree $\\Delta$ has correspondence chromatic ⏎ number $O_r(\\Delta/\\log\\Delta)$. This resolves the Alon--Krivelevich--Sudakov ⏎ coloring conjecture in the stronger correspondence-coloring form. The ⏎ same bound, with a constant depending on $F$, holds when any fixed graph ⏎ $F$ is excluded as an ordinary subgraph. Ordinary and list coloring ⏎ satisfy the same bounds.", "lean/docs/184.md:5: - [A logarithmic independence bound for clique-free graphs](../../preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf)", "lean/docs/184.md:9: The formalized result gives a logarithmic improvement in the independence number of clique-free graphs. For every fixed integer $r\\ge4$, there is $c_r>0$ such that every finite $K_r$-free simple graph on $n$ vertices with average degree $d\\ge2$ has an independent set of size at least $c_r n\\log d/d$. The constant depends only on $r$.", "lean/ComparatorChallenges/CliqueFreeLog.lean:11-16: theorem logarithmic_independence_bound (r : ℕ) (hr : 4 ≤ r) : ⏎ ∃ c : ℝ, 0 < c ∧ ⏎ ∀ {V : Type} [Fintype V] (G : SimpleGraph V), ⏎ G.CliqueFree r → 2 ≤ averageDegree G → ⏎ c * (Fintype.card V : ℝ) * Real.log (averageDegree G) / averageDegree G ≤ ⏎ (G.indepNum : ℝ) := by" ] }, { "clause": "Also proves the Ajtai–Erdős–Komlós–Szemerédi independence conjecture: for fixed r ≥ 4, every n-vertex Kr-free graph of average degree d ≥ 2 has an independent set of size $`\\Omega_r(n\\log d/d)`$.", "status": "stated", "coveredBy": [ "CliqueFreeLog" ], "encoding": "average degree 2|E|/|V| (|V| = 0 gives 0, excluded by d ≥ 2); V : Type with Fintype", "evidence": [ "lean/ComparatorChallenges/CliqueFreeLog.lean:11-16: theorem logarithmic_independence_bound (r : ℕ) (hr : 4 ≤ r) : ⏎ ∃ c : ℝ, 0 < c ∧ ⏎ ∀ {V : Type} [Fintype V] (G : SimpleGraph V), ⏎ G.CliqueFree r → 2 ≤ averageDegree G → ⏎ c * (Fintype.card V : ℝ) * Real.log (averageDegree G) / averageDegree G ≤ ⏎ (G.indepNum : ℝ) := by" ] } ], "gap": "The correspondence-coloring theorem (χ_DP = O_F(Δ/log Δ) for F-free graphs of large maximum degree), the headline's first clause, is not stated.", "scopeNoteOnGap": "silent — the note links only the independence paper: lean/docs/184.md:9: The formalized result gives a logarithmic improvement in the independence number of clique-free graphs.", "note": "The independence bound is stated with the right quantifier order (∀ r ≥ 4 ∃ c > 0 ∀ G). The correspondence-colouring theorem, the headline's first claim and the family title, is the October-5 paper, which the note does not link. Differs from the first reader only by amendment 3.", "firstSuspicious": "Pattern seen across this batch: the CONTENTS.md headline was widened to manuscripts the scope note does not link, and the scope note never says those manuscripts are unformalized. Strictly that is silence, hence UNDECLARED; a second reader should weigh whether the scope note's paper list counts as a declaration. Here: the October 5 correspondence-coloring paper.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json", "second/second6.json" ] }, { "family": "185", "title": "Counterexamples to infinite matroid intersection and packing/covering", "headline": "Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó’s partitional-matroid question negatively. They are neither finitary nor cofinitary, so Nash-Williams’ original finitary conjecture remains outside the result.", "challenges": [ "InfiniteMatroid", "InfiniteMatroidCorollaries" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set", "coveredBy": [ "InfiniteMatroid", "InfiniteMatroidCorollaries" ], "note": "E = ℤ × Σ m, ((Fin m → Bool) → Bool) is countable and infinite; ∃ self-dual M₀, M₁ (Mathlib infinite matroids) with ground set E, no independent cover I₀ ∪ I₁ = E, ¬HasPackingCovering (no partition E = P ⊔ C with disjoint spanning sets of M_i|P and covering independent sets of the contractions M_i.C), and — for the explicit rayMatroid pair, shown partitional — ¬HasIntersection (no common independent J = J₀ ⊔ J₁ with cl_{M₀}(J₀) ∪ cl_{M₁}(J₁) = E). Proofs use only propext, Quot.sound, Classical.choice." }, { "clause": "The same examples answer Joó’s partitional-matroid question negatively", "coveredBy": [ "InfiniteMatroidCorollaries" ], "note": "IsPartitional (isomorphic to a direct sum of uniform matroids, uniform = every set independent or spanning) for both, together with ¬HasIntersection and ¬HasPackingCovering." }, { "clause": "They are neither finitary nor cofinitary, so Nash-Williams’ original finitary conjecture remains outside the result", "coveredBy": [], "note": "A caveat that limits the result, not a claimed achievement: no challenge states that M₀, M₁ are non-finitary/non-cofinitary. Nothing the headline claims as proved is missing." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": "The headline's 'neither finitary nor cofinitary' is not stated by any challenge; I treated it as a disclaimer (it narrows what the result claims), not as an uncovered claim. A second reader may prefer to record it as an uncovered descriptive clause.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "186", "title": "Uniform influence and sharp thresholds for graph and hypergraph properties", "headline": "Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed $`0\\lt \\varepsilon\\lt 1/2`$, every nontrivial increasing relabeling-invariant property crosses from probability ε to $`1-\\varepsilon`$ within width $`O((\\log n)^{-2})`$ for graphs and $`O_r((\\log n)^{-r/(r-1)})`$ for r-uniform hypergraphs, r ≥ 3. The hypergraph influence bound also applies to nonmonotone properties.", "challenges": [ "SharpThreshold" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves the Friedgut–Kalai threshold-width conjectures for graphs", "status": "stated", "coveredBy": [ "SharpThreshold" ], "evidence": [ "lean/ComparatorChallenges/SharpThreshold.lean:53: 2 ≤ n → VertexInvariant f → IncreasingGraphProperty f →", "lean/ComparatorChallenges/SharpThreshold.lean:55: graphQuantile (1 - ε) f - graphQuantile ε f ≤", "lean/ComparatorChallenges/SharpThreshold.lean:56: ((2 : ℝ) ^ 19 / (Real.log (n : ℝ)) ^ 2) * Real.log (1 / (2 * ε)) := by" ] }, { "clause": "and fixed-uniformity hypergraphs.", "status": "unlinked", "coveredBy": [], "manuscript": "A-uniform-influence-bound-for-hypergraph-properties-October-5-2026", "evidence": [ "CONTENTS.md:4309: [A uniform influence bound for hypergraph properties]", "CONTENTS.md:4311: For every fixed integer r ≥ 3, we prove that every relabeling-invariant Boolean property of simple r-uniform hypergraphs on n vertices satisfies", "lean/docs/186.md:5: - [A Sharp Threshold Bound for Monotone Graph Properties](../../preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf)" ] }, { "clause": "For fixed $`0\\lt \\varepsilon\\lt 1/2`$, every nontrivial increasing relabeling-invariant property crosses from probability ε to $`1-\\varepsilon`$ within width $`O((\\log n)^{-2})`$ for graphs", "status": "stated", "coveredBy": [ "SharpThreshold" ], "evidence": [ "lean/ComparatorChallenges/SharpThreshold.lean:54: NontrivialGraphProperty f → 0 < ε → ε < (1 / 2 : ℝ) →", "lean/ComparatorChallenges/SharpThreshold.lean:49: sInf {p : ℝ | p ∈ Set.Icc (0 : ℝ) 1 ∧ graphMean p f ≥ a}" ], "note": "Stronger than O((log n)^{-2}) for fixed ε: the explicit bound 2^19·log(1/(2ε))/(log n)^2 for every n ≥ 2. The quantile is an sInf over [0,1] of a nonempty set for a nontrivial increasing property (mean 0 at p=0, 1 at p=1), so no junk value." }, { "clause": "and $`O_r((\\log n)^{-r/(r-1)})`$ for r-uniform hypergraphs, r ≥ 3.", "status": "unlinked", "coveredBy": [], "manuscript": "A-uniform-influence-bound-for-hypergraph-properties-October-5-2026", "evidence": [ "CONTENTS.md:4309: [A uniform influence bound for hypergraph properties]", "CONTENTS.md:4311: For every fixed integer r ≥ 3, we prove that every relabeling-invariant Boolean property of simple r-uniform hypergraphs on n vertices satisfies", "CONTENTS.md:4311: For increasing properties, it gives the corresponding threshold-width bound with exponent $`r/(r-1)`$, proving the hypergraph threshold-width conjecture of Friedgut and Kalai.", "lean/docs/186.md:5: - [A Sharp Threshold Bound for Monotone Graph Properties](../../preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf)" ] }, { "clause": "The hypergraph influence bound also applies to nonmonotone properties.", "status": "unlinked", "coveredBy": [], "manuscript": "A-uniform-influence-bound-for-hypergraph-properties-October-5-2026", "evidence": [ "CONTENTS.md:4311: the bound holds uniformly for all $`0\\lt p\\lt 1`$ without a monotonicity assumption.", "lean/docs/186.md:5: - [A Sharp Threshold Bound for Monotone Graph Properties](../../preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf)" ] } ], "gap": "The whole hypergraph half (threshold width O_r((log n)^{-r/(r−1)}) for r ≥ 3, and the Var ≤ C_r·I/(log n)^{r/(r−1)} influence bound for nonmonotone properties) — the October 5 manuscript, which the scope note does not link.", "scopeNoteOnGap": "silent: the note links only the September 25 graph manuscript (186.md:5) and describes only the graph theorem (186.md:9); hypergraphs are not mentioned in the Scope section", "note": "The graph conjecture is stated with an explicit constant; the whole hypergraph half (threshold width for r ≥ 3 and the nonmonotone influence bound) comes from the October-5 manuscript docs/186.md does not link (it links only the September-25 graph paper): NARROWER — UNLINKED. The first reader found the same gap and called it UNDECLARED before amendment 3; the difference is the amendment.", "firstSuspicious": "Same pattern as 084, 197, 247 (headline extended by an unlinked October manuscript).", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch6.json", "second/second7.json" ] }, { "family": "187", "title": "Snaky in 21 Maker moves", "headline": "Settles the Snaky achievement problem: Maker can force the six-cell Snaky shape within 21 of its own moves on the initially empty infinite square board. Maker moves first, each player claims one free cell per turn, and translations, rotations and reflections count as wins.", "challenges": [ "SnakyCertificate", "SnakyConditional", "SnakyTwentyOne" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Maker can force the six-cell Snaky shape within 21 of its own moves on the initially empty infinite square board.", "coveredBy": [ "SnakyTwentyOne" ], "note": "∃ legal Maker policy that, against every legal Breaker sequence from ∅, ∅ on Z², holds a placed copy of S after 21 Maker moves." }, { "clause": "Maker moves first, each player claims one free cell per turn, and translations, rotations and reflections count as wins.", "coveredBy": [ "SnakyTwentyOne" ], "note": "Rounds: Maker plays σ(state) then Breaker β_k; both must take free cells; HasSnaky allows any of 8 orientations and any translation." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": "SnakyConditional's statement does not match its scope-note description (conditional templates vs an unconditional 35-move win). Harmless for the family word; worth a second reader and a line on the page.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "SnakyConditional", "flag": "SCOPE-NOTE MISMATCH: docs/187.md:11 describes the supplements as proving 'four conditional winning templates from partial positions, assuming the required Maker cells are present and Breaker avoids the corresponding finite envelope', and the link table calls SnakyConditional 'Conditional winning templates from finite partial positions', but the Comparator statement is an unconditional 35-move win from M₀ = B₀ = ∅ with no template or envelope hypothesis" }, { "challenge": "SnakyCertificate", "flag": "LARGE STATEMENT: 1,399 lines. Closure read: Cell/snaky/orient/placement, Term/Entry, entry_6 … entry_615, tableChunk_0..30, table, entryAt, TableValid, placementCount, RowData/baseRow/placeRow/unionA/unionH/commonH/forkRow/rowAt, Output, AllRows — all concrete data; it cannot carry the headline either way" } ], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "188", "title": "The sharp terminal leave in random triangle removal", "headline": "Starting from the complete graph on n vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to $`n^{3/2}/(2\\sqrt2)`$, with mean-square convergence after normalization by n3/2. This proves the triangle case of the Joos–Kühn sharp-constant conjecture.", "challenges": [ "TriangleRemoval" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Starting from the complete graph on n vertices, repeatedly delete a uniformly chosen remaining triangle.", "coveredBy": [ "TriangleRemoval" ], "note": "step: uniform triangle among those of G, remove its 3 edges; evolve for C(n,2) steps (enough to terminate)." }, { "clause": "The terminal edge count is asymptotic to $`n^{3/2}/(2\\sqrt2)`$, with mean-square convergence after normalization by n3/2.", "coveredBy": [ "TriangleRemoval" ], "note": "E[(|E_final|/n^{3/2} − 1/(2√2))²] → 0, plus convergence in probability and of the normalized expectation." }, { "clause": "This proves the triangle case of the Joos–Kühn sharp-constant conjecture.", "coveredBy": [ "TriangleRemoval" ], "note": "Same statement." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch5.json" ] }, { "family": "189", "title": "Cycle–clique Ramsey numbers", "headline": "Proves the Erdős–Faudree–Rousseau–Schelp conjecture: $`R(C_m,K_n)=(m-1)(n-1)+1`$ for every $`m\\ge n\\ge3`$, except $`R(C_3,K_3)=6`$. This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph.", "challenges": [ "CycleCliqueRamsey" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "$`R(C_m,K_n)=(m-1)(n-1)+1`$ for every $`m\\ge n\\ge3`$, except $`R(C_3,K_3)=6`$", "coveredBy": [ "CycleCliqueRamsey" ], "note": "∀ integers m ≥ n ≥ 3, (m,n) ≠ (3,3): least N with the Ramsey property = (m−1)(n−1)+1; and R(C₃,K₃) = 6." }, { "clause": "This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph.", "coveredBy": [ "CycleCliqueRamsey" ], "note": "RamseyProperty: every graph G on Fin N contains cycleGraph m as a subgraph or its complement contains K_n." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "190", "title": "Polynomial removal fails for ordered binary matrices", "headline": "Disproves polynomial ordered binary matrix removal with one fixed $`66\\times66`$ zero–one pattern. Matrices can require many binary-entry changes to become pattern-free while their copy density is smaller than every proposed polynomial bound in that distance. Copies preserve row and column orders and match both zeros and ones.", "challenges": [ "MatrixRemoval" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Disproves polynomial ordered binary matrix removal with one fixed $`66\\times66`$ zero–one pattern.", "coveredBy": [ "MatrixRemoval" ], "note": "fixedH explicit 66×66." }, { "clause": "Matrices can require many binary-entry changes to become pattern-free while their copy density is smaller than every proposed polynomial bound in that distance.", "coveredBy": [ "MatrixRemoval" ], "note": "∀ c, C > 0 ∃ n, ε ∈ (0,1), A: dist(A, H-free) ≥ ε·n² edits yet copies < c·ε^C·n^132 — the exact negation of ∃ c, C ∀ n, ε, A." }, { "clause": "Copies preserve row and column orders and match both zeros and ones.", "coveredBy": [ "MatrixRemoval" ], "note": "Copies = pairs of strictly increasing row and column maps with A(r i, c j) = H i j for all i, j (both values matched)." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "MatrixRemoval", "flag": "JUNK checked, none: fixedMinEdits uses the sentinel n²+1 for non-free B, but an H-free matrix always exists (the all-zero matrix, since fixedH contains a 1), so the minimum is the true edit distance; for n < 66 every matrix is free, the distance is 0 and the claim cannot be met trivially" } ], "flagReadings": null, "readings": [ "first/batch6.json" ] }, { "family": "191", "title": "A power improvement in the Heilbronn triangle lower bound", "headline": "For every sufficiently large n, constructs n points in the unit square such that every triangle has area at least $`n^{-2+c}`$ for one absolute c > 0. This disproves the conjectured almost-n−2 upper bound in Heilbronn's triangle problem, which asks how large the smallest determined triangle can be.", "challenges": [ "HeilbronnTriangle" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "For every sufficiently large n, constructs n points in the unit square such that every triangle has area at least $`n^{-2+c}`$ for one absolute c > 0.", "coveredBy": [ "HeilbronnTriangle" ], "note": "Only along an unbounded sequence of sizes n_j → ∞ (∃ n : ℕ → ℕ tending to ∞), with the explicit exponent η = 1/(100000·K)." }, { "clause": "This disproves the conjectured almost-n−2 upper bound in Heilbronn's triangle problem", "coveredBy": [ "HeilbronnTriangle" ], "note": "almost_n_minus_two_refuted: the bound '∃C ∃n₀ ∀n ≥ n₀ every n-point set has a triangle of area ≤ C n^{−2+ε}' fails for ε = η/2 (a sequence suffices for this)." } ], "gap": "Only an unbounded sequence of sizes n_j is stated, not every sufficiently large n.", "scopeNoteOnGap": "lean/docs/191.md:9: The linked construction is for an unbounded sequence of sizes; the paper's statement for every sufficiently large $n$ is broader.", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "HeilbronnTriangle", "flag": "QUANTIFIER (declared): ∃ sequence n_j → ∞ instead of ∀ n ≥ n₀" } ], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "192", "title": "Boolean functions violate the square-root degree bound by arbitrary factors", "headline": "Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every C > 0, there is a sign-valued Boolean function f with $`\\sum_i\\widehat f(\\{i\\})\\gt C\\sqrt{\\deg(f)}`$. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor.", "challenges": [ "SquareRootDegree" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every C > 0, there is a sign-valued Boolean function f with $`\\sum_i\\widehat f(\\{i\\})\\gt C\\sqrt{\\deg(f)}`$. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor.", "coveredBy": [ "SquareRootDegree" ], "note": "SignedViolations: ∀ C > 0 ∃ n > 0 and a ±1-valued nonconstant f on {±1}^n with C·√deg(f) < Σ_i f̂({i}); plus unboundedness of Σ|f̂({i})|/√deg f." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch3.json" ] }, { "family": "194", "title": "Lech’s multiplicity conjecture", "headline": "Proves $`e(R)\\le e(S)`$ for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic.", "challenges": [ "DuttaDomain" ], "word": "DIFFERENT", "firstWord": "DIFFERENT", "secondWord": "DIFFERENT", "disagreement": null, "clauses": [ { "clause": "Proves $`e(R)\\le e(S)`$ for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity.", "status": "declared", "coveredBy": [], "manuscript": "Lechs-multiplicity-conjecture-September-23-2026", "evidence": [ "lean/docs/194.md:11: This selected statement is the complete-domain Dutta comparison. The paper's full flat-local result in arbitrary characteristic is outside it.", "lean/ComparatorChallenges/DuttaDomain.lean:62: def DuttaDomainClaim : Prop :=", "lean/ComparatorChallenges/DuttaDomain.lean:63: ∀ (D : Type u) [CommRing D] [IsNoetherianRing D] [IsLocalRing D] [IsDomain D]", "lean/ComparatorChallenges/DuttaDomain.lean:64: [IsAdicComplete (IsLocalRing.maximalIdeal D) D]", "lean/ComparatorChallenges/DuttaDomain.lean:65: (p : ℕ) [Fact p.Prime] [CharP D p]", "lean/ComparatorChallenges/DuttaDomain.lean:66: (F : CochainComplex (ModuleCat.{u} D) ℤ), IsShortComplex D F →", "lean/ComparatorChallenges/DuttaDomain.lean:67: (∀ n i,IsFiniteLength D ((frobeniusComplex D p n F).homology i)) ∧", "lean/ComparatorChallenges/DuttaDomain.lean:68: Tendsto (duttaSequence D p F) atTop (𝓝 (duttaMultiplicity D p F)) ∧", "lean/ComparatorChallenges/DuttaDomain.lean:69: multiplicity D ≤ duttaMultiplicity D p F" ] }, { "clause": "This resolves Lech's conjecture in every dimension and characteristic.", "status": "declared", "coveredBy": [], "manuscript": "Lechs-multiplicity-conjecture-September-23-2026", "evidence": [ "lean/docs/194.md:11: The paper's full flat-local result in arbitrary characteristic is outside it.", "lean/docs/194.md:9: The linked formalization covers a supporting characteristic-$p$ comparison over a complete Noetherian local domain $D$" ] } ], "gap": "The whole headline. The challenge is a characteristic-p lemma about one ring: for a complete Noetherian local domain D of characteristic p and a 'short complex' F (finite free, concentrated in degrees [−dim D, 0], finite-length homology, H⁰ ≠ 0), the Frobenius-twisted Euler characteristics p^{−n·dim D}·χ(F^n(F)) converge (the Dutta multiplicity) and e(D) is at most that limit. It is neither Lech's inequality nor implied by it.", "scopeNoteOnGap": "lean/docs/194.md:11: \"This selected statement is the complete-domain Dutta comparison. The paper's full flat-local result in arbitrary characteristic is outside it.\"; 194.md:9 calls it \"a supporting characteristic-$p$ comparison\".", "note": "No clause is stated: DuttaDomain is a char-p comparison e(D) ≤ Dutta multiplicity for one complete local domain, which neither implies Lech's inequality nor is implied by it, so DIFFERENT by the precedence rule — although every clause is declared (194.md:11) and the note itself calls the statement supporting. It is not on the README's supporting list (README.md:17-26), so not SUPPORT-ONLY. Agree with the first reader.", "firstSuspicious": "DIFFERENT vs SUPPORT-ONLY: OpenAI's own scope note calls it supporting; the README list does not include it. Reclassify uniformly with 033 and 281 if scope-note declarations are to count as SUPPORT.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "DuttaDomain", "flag": "JUNK (low): multiplicity is Filter.limUnder, a junk value if the limit did not exist; for Noetherian local rings the Hilbert–Samuel limit exists, so the value is genuine; the Dutta limit is asserted by Tendsto, not assumed" }, { "challenge": "DuttaDomain", "flag": "SUPPORT? declared supporting in the scope note (194.md:9) but not listed among the supporting-result comparisons in ComparatorChallenges/README.md; by the pre-registered definition (README) the word is DIFFERENT" } ], "flagReadings": null, "readings": [ "first/batch6.json", "second/second1.json" ] }, { "family": "195", "title": "A counterexample to the small Cohen–Macaulay module conjecture", "headline": "Constructs a three-dimensional complete Noetherian normal local domain over ℂ with no nonzero finitely generated maximal Cohen–Macaulay module. A three-dimensional local domain essentially of finite type over ℂ has the same property, disproving the domain form of the small Cohen–Macaulay module conjecture.", "challenges": [ "SurfaceConeCandidateSupport" ], "word": "SUPPORT-ONLY", "firstWord": "SUPPORT-ONLY", "secondWord": "SUPPORT-ONLY", "disagreement": null, "clauses": [ { "clause": "Constructs a three-dimensional complete Noetherian normal local domain over ℂ with no nonzero finitely generated maximal Cohen–Macaulay module.", "status": "declared", "coveredBy": [], "manuscript": "A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026", "evidence": [ "lean/docs/195.md:9: The linked formalization verifies only the ring-theoretic candidate", "lean/docs/195.md:11: It does not prove that this ring lacks a nonzero finitely generated maximal Cohen–Macaulay module, or the paper's counterexample essentially of finite type.", "lean/ComparatorChallenges/README.md:21: - `SurfaceConeCandidateSupport.json`: ring-theoretic properties of the completed" ], "note": "The support theorem asserts only that SOME 3-dimensional complete Noetherian integrally closed local ℂ-domain with residue field ℂ exists. ℂ[[x,y,z]] witnesses it, so it carries nothing specific to the counterexample. OpenAI says so: it is declared support." }, { "clause": "A three-dimensional local domain essentially of finite type over ℂ has the same property, disproving the domain form of the small Cohen–Macaulay module conjecture.", "status": "declared", "coveredBy": [], "manuscript": "A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026", "evidence": [ "lean/docs/195.md:11: or the paper's counterexample essentially of finite type." ], "note": "Declared in the same sentence." } ], "gap": "The counterexample itself (no nonzero finitely generated MCM module) and the essentially-of-finite-type example are not stated; only properties of the candidate ring are, and those properties are existential.", "scopeNoteOnGap": "docs/195.md:9: …The linked formalization verifies only the ring-theoretic candidate… docs/195.md:11: It does not prove that this ring lacks a nonzero finitely generated maximal Cohen–Macaulay module, or the paper's counterexample essentially of finite type.", "note": "The family's only challenge is on the README's list of supporting results, and the note names both omissions, so the family is SUPPORT-ONLY with every clause DECLARED. Agrees with the first reader.", "firstSuspicious": "The support statement is satisfied by the power-series ring ℂ[[x,y,z]], so as written it verifies nothing about the paper's construction (OpenAI declares it support-only and says so in docs/195.md).", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "SurfaceConeCandidateSupport", "flag": "VACUITY?: the statement is purely existential and does not name the paper's ring, so it is witnessed by ℂ[[x,y,z]] (complete, Noetherian, regular hence integrally closed, local, dimension 3, residue field ℂ) — it carries no information specific to the counterexample." } ], "flagReadings": null, "readings": [ "first/batch2.json", "second/second2.json" ] }, { "family": "196", "title": "A counterexample to Kaplansky’s zero-divisor conjecture", "headline": "Constructs a finitely presented torsion-free group G whose group algebra $`\\mathbb F_2[G]`$ has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.", "challenges": [ "TorsionFreeZeroDivisors" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a finitely presented torsion-free group G whose group algebra $`\\mathbb F_2[G]`$ has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture", "coveredBy": [ "TorsionFreeZeroDivisors" ], "note": "There exists G: finitely presented, torsion-free, with alpha, beta != 0 in MonoidAlgebra (ZMod 2) G and alpha * beta = 0." }, { "clause": "The group has a finite two-dimensional classifying space", "coveredBy": [ "TorsionFreeZeroDivisors" ], "note": "A finite, Hausdorff, path-connected CW complex with no cells above dimension 2 and at least one 2-cell, pi_1 isomorphic to G, and a contractible surjective covering space (so it is aspherical)." } ], "gap": null, "scopeNoteOnGap": "no gap", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "197", "title": "A torsion-free group algebra that is not directly finite", "headline": "Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽2 is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.", "challenges": [ "KaplanskyFinitelyPresented", "KaplanskyDirectFiniteness", "GroupRingDeterminant", "OddKaplansky" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Constructs a finitely presented … group whose group algebra … is not directly finite, disproving Kaplansky's conjecture", "status": "stated", "coveredBy": [ "KaplanskyFinitelyPresented", "KaplanskyDirectFiniteness" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/KaplanskyFinitelyPresented.lean:8: ∃ (K : Type) (_ : Field K) (_ : Fintype K) (_ : CharP K 2),", "lean/ComparatorChallenges/KaplanskyFinitelyPresented.lean:9: ∃ (G : Type) (_ : Group G) (_ : Group.IsFinitelyPresented G),", "lean/ComparatorChallenges/KaplanskyFinitelyPresented.lean:11: ∃ a_out b_out : MonoidAlgebra K G, a_out * b_out = 1 ∧ b_out * a_out ≠ 1 := by" ] }, { "clause": "torsion-free", "status": "unlinked", "coveredBy": [], "manuscript": "A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026", "evidence": [ "lean/ComparatorChallenges/KaplanskyFinitelyPresented.lean:10: (∃ (ℓ : ℕ) (_ : ℓ.Prime) (_ : Odd ℓ) (g : G), orderOf g = ℓ) ∧", "lean/docs/197.md:11: the more detailed construction gives a finitely presented group with an element of odd prime order.", "preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/build/sections/introduction.tex:14: There exist a finitely presented torsion-free group $G$ and elements", "preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/build/sections/01-introduction.tex:18: The construction uses odd-order torsion and does not assert that $K=\\F_2$.", "lean/docs/197.md:1: # A torsion-free group algebra that is not directly finite", "lean/docs/197.md:5: - [A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two]", "lean/docs/197.md:6: - [A Counterexample to the Group-Ring Determinant Conjecture]", "lean/docs/197.md:7: - [A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic]" ] }, { "clause": "nonsofic", "status": "declared", "coveredBy": [], "manuscript": "A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026", "evidence": [ "lean/docs/197.md:13: The further conclusion that the group is nonsofic is outside these statements." ] }, { "clause": "over 𝔽2", "status": "unlinked", "coveredBy": [], "manuscript": "A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026", "evidence": [ "lean/ComparatorChallenges/KaplanskyFinitelyPresented.lean:8: ∃ (K : Type) (_ : Field K) (_ : Fintype K) (_ : CharP K 2),", "preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/build/sections/introduction.tex:49: Theorem~\\ref{thm:main} uses the prime field $\\Ftwo$ and a torsion-free group.", "preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/build/sections/01-introduction.tex:18: does not assert that $K=\\F_2$" ] }, { "clause": "even without torsion", "status": "unlinked", "coveredBy": [], "manuscript": "A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026", "evidence": [ "preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/build/sections/introduction.tex:9: group is finitely presented and torsion-free.", "lean/ComparatorChallenges/KaplanskyFinitelyPresented.lean:10: (∃ (ℓ : ℕ) (_ : ℓ.Prime) (_ : Odd ℓ) (g : G), orderOf g = ℓ) ∧" ] }, { "clause": "Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture.", "status": "stated", "coveredBy": [ "OddKaplansky" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/OddKaplansky.lean:14: d.coeff.sum fun u c => c * x (g * u)", "lean/ComparatorChallenges/OddKaplansky.lean:18: ∃ (K : Type) (_ : Field K) (_ : Fintype K) (_ : CharP K sourcePrime),", "lean/ComparatorChallenges/OddKaplansky.lean:24: Function.Injective (cellular b) ∧ ¬ Function.Surjective (cellular b)" ] }, { "clause": "Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.", "status": "stated", "coveredBy": [ "GroupRingDeterminant" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/GroupRingDeterminant.lean:28: Real.exp ((groupTrace (CFC.log (star T * T))).re / 2)", "lean/ComparatorChallenges/GroupRingDeterminant.lean:49: IsUnit (rationalMatrix A) ∧ IsRegularOperator A T ∧ IsUnit T ∧", "lean/ComparatorChallenges/GroupRingDeterminant.lean:50: 0 < fkDet T ∧ fkDet T < 1" ] } ], "gap": "The headline's lead claim — a TORSION-FREE, nonsofic, finitely presented group with 𝔽2[G] not directly finite — is not stated. The finitely presented Lean example is required to have torsion (∃ g with orderOf g = an odd prime) and lives over an unspecified finite field of characteristic 2. Nonsoficity is declared outside; torsion-freeness and the field 𝔽2 are not mentioned by the scope note, which does not link the October 4 torsion-free manuscript.", "scopeNoteOnGap": "partly declared: lean/docs/197.md:13: \"The further conclusion that the group is nonsofic is outside these statements.\"; silent on torsion-freeness and on 𝔽2 — the note describes lean/docs/197.md:11: \"the more detailed construction gives a finitely presented group with an element of odd prime order\" and links only the three September manuscripts (197.md:5–7), not 'A Torsion-Free Group Algebra That Is Not Directly Finite' (October 4)", "note": "The five clause entries above partition the headline's first sentence. The finitely presented Lean example is REQUIRED to carry an element of odd prime order (KaplanskyFinitelyPresented.lean:10), over an unspecified finite field of characteristic 2; no challenge states a torsion-free group or the field 𝔽2 (passing from F_{2^k} to 𝔽2 would need the block decomposition of 𝔽2[C_{2^k−1}] plus a corner argument, not formalized, and still gives a group with torsion). Those clauses come only from the October 4 manuscript, which the note does not link; the linked September 23 paper says in words that it uses odd-order torsion and does not assert K = 𝔽2. Nonsoficity is declared (197.md:13). Gottschalk is stated by OddKaplansky (a finite-alphabet linear cellular automaton, G-equivariant with finite memory supp b; the characteristic-two example's automaton is not separately stated, but one stated example carries the clause). The determinant clause is stated (FK determinant of an invertible operator as exp(½ tr log T*T); left-regular convention, equivalent under g ↦ g⁻¹ and transposition; no junk: a missing CFC instance would give fkDet = 1, excluded by fkDet < 1). Word NARROWER — UNLINKED; I differ from the first reader by the amendment-3 word only. The first reader's facts are confirmed, and one is sharper here: the scope note's own heading (197.md:1) is the torsion-free title, sitting above a body that says the formal group has odd-prime-order torsion (197.md:11). A heading does not declare an omission, but a reader who stops at it is told the opposite of what the statements cover.", "firstSuspicious": "Strongest finding in the batch: the family title and lead sentence are 'torsion-free', and the formal finitely presented example explicitly has torsion. Second reading required by the discipline.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "GroupRingDeterminant", "flag": "JUNK checked, none: fkDet uses CFC.log (star T * T); for a non-invertible T the continuous functional calculus would return 0 and give fkDet = 1, outside (0,1), and IsUnit T is asserted — no trivialization" } ], "flagReadings": null, "readings": [ "first/batch6.json", "second/second3.json" ] }, { "family": "198", "title": "A counterexample to finitistic-dimension finiteness", "headline": "Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.", "challenges": [ "LittleFinitistic", "FinitisticAsymmetry" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions", "coveredBy": [ "LittleFinitistic", "FinitisticAsymmetry" ], "note": "LittleFinitistic: ∃ a ring A with a ℂ-algebra structure, finite-dimensional over ℂ, whose little finitistic dimension (⨆ of projectiveDimension over finitely generated A-modules of finite projective dimension, in WithBot ℕ∞) is ⊤, and for every m ≥ 1 a finitely generated module N with 2m−2 ≤ pd N < ⊤. Over a finite-dimensional algebra, finitely generated = finite-dimensional." }, { "clause": "This disproves the little finitistic-dimension conjecture", "coveredBy": [ "LittleFinitistic", "FinitisticAsymmetry" ], "note": "littleFinitisticDimension A = ⊤." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "199", "title": "Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures", "headline": "Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander–Reiten and Gorenstein-projective conjectures, and Tachikawa's second conjecture. An associated endomorphism algebra also disproves the classical, generalized and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. The counterexamples persist under every extension of the base field.", "challenges": [ "AuslanderReiten", "Tachikawa" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander–Reiten and Gorenstein-projective conjectures, and Tachikawa's second conjecture.", "coveredBy": [ "AuslanderReiten", "Tachikawa" ], "note": "AR: f.d. algebra over F₂(q,H₁,H₂) with f.d. nonprojective Z, Ext^i(Z,Z) = Ext^i(Z,R) = 0 for i > 0, Z Gorenstein projective (totally acyclic witness). Tachikawa: f.d. symmetric algebra with f.d. nonprojective M, Ext^i(M,M) = 0 for i > 0." }, { "clause": "An associated endomorphism algebra also disproves the classical, generalized and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture.", "coveredBy": [], "note": "Not stated (declared)." }, { "clause": "The counterexamples persist under every extension of the base field.", "coveredBy": [ "AuslanderReiten" ], "note": "Stated for the AR/Gorenstein counterexample (∀ field extensions E of K, the Conclusions hold over E). Not stated for the Tachikawa counterexample (declared)." } ], "gap": "The endomorphism-algebra counterexamples (three Nakayama conjectures, Auslander–Gorenstein, Wakamatsu tilting) and field-extension persistence of the Tachikawa counterexample are not stated.", "scopeNoteOnGap": "lean/docs/199.md:16: The later field-extension, endomorphism-algebra, and related homological consequences are not included.", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "205", "title": "Saxl’s conjecture and universal tensor squares", "headline": "Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every Sn with $`n\\notin\\{2,4,9\\}`$ has an irreducible representation whose tensor square contains all irreducibles.", "challenges": [ "UniversalTensorSquares", "Saxl" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group.", "coveredBy": [ "Saxl" ], "note": "for every m>=1 and every μ with |μ| = m(m+1)/2, the Kronecker multiplicity g(ρ_m,ρ_m,μ) = dim Hom_{S_n}(S^μ, S^ρ ⊗ S^ρ) > 0, with Specht modules built explicitly" }, { "clause": "More generally, every Sn with $`n\\notin\\{2,4,9\\}`$ has an irreducible representation whose tensor square contains all irreducibles.", "coveredBy": [ "UniversalTensorSquares" ], "note": "for n>0 (S_0 trivial), not 2,4,9; also directly: every finite-dimensional irreducible ρ of S_n injects into S^λ ⊗ S^λ, and S^λ is asserted irreducible" } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch8.json" ] }, { "family": "206", "title": "Finite lattice representation and undecidability", "headline": "Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively. Moreover, no algorithm decides whether a finite lattice has such a representation, or whether it is a full subgroup interval of a finite group.", "challenges": [ "FiniteCongruenceGraph" ], "word": "DIFFERENT", "firstWord": "DIFFERENT", "secondWord": "DIFFERENT", "disagreement": null, "clauses": [ { "clause": "Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively.", "status": "declared", "coveredBy": [], "manuscript": "Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026 (also, and primarily, the unlinked A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026)", "evidence": [ "CONTENTS.md:4707: and prove that deciding this representation property is undecidable. In particular, the finite lattice representation problem has a negative answer.", "preprints/Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026/build/sections/introduction.tex:78: Part~\\textup{(ii)} resolves the finite lattice representation problem", "lean/docs/206.md:9: This selected theorem is the equivalence with the graph criterion; the paper's undecidability and subgroup-interval conclusions are outside it.", "CONTENTS.md:4716: We prove that there is a finite nonempty lattice that is not the full congruence lattice of any finite nonempty algebra of any finite signature.", "lean/ComparatorChallenges/FiniteCongruenceGraph.lean:58: Representable Lat ↔ HasGraphWitness Lat := by" ], "why": "Not stated: the criterion exhibits no lattice without a graph witness. In the linked paper the negative answer is drawn from the undecidability ('Part (ii) resolves ... negatively'; the abstract's 'In particular'), which the note names as outside; I read that as declaring it. Its explicit minimum non-representable lattice (Theorem part (iii)) is not separately named, and the clause's other source, the Negative-Solution paper, is unlinked; a stricter reader would write undeclared or unlinked. The family word does not depend on this." }, { "clause": "Moreover, no algorithm decides whether a finite lattice has such a representation, or whether it is a full subgroup interval of a finite group.", "status": "declared", "coveredBy": [], "manuscript": "Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026", "evidence": [ "lean/docs/206.md:9: This selected theorem is the equivalence with the graph criterion; the paper's undecidability and subgroup-interval conclusions are outside it." ] } ], "gap": "every clause of the headline: existence of a non-representable finite lattice, undecidability of representability, undecidability of subgroup-interval recognition. The Lean states instead a characterization (representable ⇔ colored-graph witness) that neither implies nor is implied by any headline clause.", "scopeNoteOnGap": "This selected theorem is the equivalence with the graph criterion; the paper's undecidability and subgroup-interval conclusions are outside it. (docs/206.md:9) — declares the undecidability clauses out; silent on the first clause (existence of a non-representable lattice) except insofar as the first manuscript derives it 'in particular' from undecidability", "note": "No headline clause is stated, and the one statement, the colored-graph characterization (representable ⇔ graph witness), neither implies nor is implied by non-representability or by either undecidability claim, so DIFFERENT by the pre-registered definition. Agree with the first reader. The note does name the omissions, so this DIFFERENT is not a silence: the headline announces the paper's corollaries and the formalization chose its characterization theorem.", "firstSuspicious": "DIFFERENT follows the pre-registered definition (neither implies nor is implied); the scope note openly declares the undecidability clauses out, so the row should say 'declared for two of three clauses' when stated on the page. Read a second time per the discipline.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "FiniteCongruenceGraph", "flag": "VACUITY? cleared: Lat has OrderBot so it is nonempty; the witness conditions are not contradictory (the trivial lattice has the 1-point witness)" } ], "flagReadings": null, "readings": [ "first/batch8.json", "second/second4.json" ] }, { "family": "207", "title": "The ℓ¹-Bass conjecture for all discrete groups", "headline": "Proves the ℓ1-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over $`\\ell^1(G)`$ are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain.", "challenges": [ "BassTrace", "BassTorsionFree" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Proves the ℓ1-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over $`\\ell^1(G)`$ are supported on finitely many finite-order conjugacy classes.", "status": "unlinked", "coveredBy": [], "manuscript": "The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026", "evidence": [ "lean/ComparatorChallenges/BassTrace.lean:471: (x : ModuleK0.{u,v} (MonoidAlgebra ℂ G)) :", "lean/ComparatorChallenges/BassTrace.lean:472: (∀ g : G, ¬IsOfFinOrder g → HattoriStallings x (ConjClasses.mk g) = 0) ∧", "CONTENTS.md:4732: The Hattori–Stallings trace of every idempotent matrix over the complex ℓ1 group algebra is supported on finitely many conjugacy classes of finite-order elements.", "lean/docs/207.md:1: # The ℓ¹-Bass conjecture for all discrete groups", "lean/docs/207.md:5: - [The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture](../../preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf)", "lean/docs/207.md:9: The formalization proves the complex Bass trace conjecture for every group" ] }, { "clause": "The algebraic companion proves the integral Bass trace conjecture", "status": "undeclared", "coveredBy": [], "manuscript": "The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026", "classicalTheorem": "Linnell's restriction (Linnell 1983, Lemma 4.1; Berrick–Hesselholt 2015, Theorem A) for the nonidentity finite-order classes, plus base change K₀(ℤG) → K₀(ℂG) — neither formalized; K₀(ℤG) occurs in no challenge.", "evidence": [ "lean/ComparatorChallenges/BassTrace.lean:467: def HattoriStallings : ModuleK0.{u,v} (MonoidAlgebra ℂ G) →+ (ConjClasses G →₀ ℂ) :=", "preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/09-completion.tex:51: The integral", "preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/09-completion.tex:52: Bass trace conjecture asks for support only at the identity class.", "preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/09-completion.tex:55: For every discrete group $G$, the Hattori--Stallings trace of every", "preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/09-completion.tex:56: class in $K_0(\\Z G)$ is supported on $[1]_G$.", "preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/09-completion.tex:68: For finite-order elements we use Linnell's restriction", "lean/docs/207.md:5: - [The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture](../../preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf)", "lean/docs/207.md:9: the Hattori–Stallings trace of each virtual class of finitely generated projective right $\\mathbb C[G]$-modules vanishes on conjugacy classes of infinite-order elements.", "lean/docs/207.md:11: it identifies the trace on $K_0(\\mathbb C[G])$ with the integral augmentation rank" ] }, { "clause": "and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain.", "status": "stated", "coveredBy": [ "BassTorsionFree" ], "reading": "torsion_free_corollary, second conjunct: for TorsionFree G and every CommRing R with IsDomain and CharZero, e² = e in R[G] ⇒ e = 0 ∨ e = 1.", "evidence": [ "lean/ComparatorChallenges/BassTorsionFree.lean:299: theorem torsion_free_corollary (G : Type*) [Group G] (hG : TorsionFree G) :", "lean/ComparatorChallenges/BassTorsionFree.lean:305: (∀ (R : Type*) [CommRing R] [IsDomain R] [CharZero R]", "lean/ComparatorChallenges/BassTorsionFree.lean:306: (e : MonoidAlgebra R G), e ^ 2 = e → e = 0 ∨ e = 1) := by" ] } ], "gap": "The ℓ¹(G) version (the family's titular result and the October 5 paper) and the integral Bass trace conjecture for K₀(ℤG) are not stated. Only the ℂ[G] Bass conjecture and the torsion-free consequences are.", "scopeNoteOnGap": "silent on ℓ¹(G) and on K₀(ℤG). ℓ¹ appears only in the title (lean/docs/207.md:1: \"# The ℓ¹-Bass conjecture for all discrete groups\"). The scope text (lines 9, 11) describes the ℂ[G] result and the torsion-free consequences, and only the ℂ[G] paper is linked (line 5).", "note": "Stated: the complex group-ring Bass trace conjecture (not in the headline), trace = augmentation rank for torsion-free G, and characteristic-zero Kaplansky. The ℓ¹(G) theorem — the family's title — is from the unlinked October-5 paper (UNLINKED); the integral Bass trace conjecture is the linked companion's Corollary cor:integral (every discrete group, K₀(ℤG)), which needs Linnell's restriction and which docs/207.md never names as left out (UNDECLARED). Reading 'for torsion-free groups' as also qualifying the integral conjecture would make that clause an elementary base change from BassTrace, but no challenge states it and the manuscript's own term is the every-group statement. Agrees with the first reader.", "firstSuspicious": "The headline (and the scope note's title) is the ℓ¹ result from the October 5 manuscript, while the formalization and the scope note's linked paper are the September 24 ℂ[G] result. The note's title promises ℓ¹ and its body never mentions it.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch5.json", "second/second5.json" ] }, { "family": "210", "title": "Foulkes' conjecture for sixth powers and quadratic stabilization", "headline": "Proves the sixth case of Foulkes’ conjecture: $`\\mathop{\\mathrm{Sym}}\\nolimits ^6(\\mathop{\\mathrm{Sym}}\\nolimits ^bV)`$ embeds equivariantly in $`\\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^6V)`$ for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map $`\\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^aV)\\to\\mathop{\\mathrm{Sym}}\\nolimits ^a(\\mathop{\\mathrm{Sym}}\\nolimits ^bV)`$ is surjective for a ≥ 2 and $`b\\ge a(a-1)`$, giving dimension-independent quadratic stabilization.", "challenges": [ "FoulkesHowe" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves the sixth case of Foulkes’ conjecture: $`\\mathop{\\mathrm{Sym}}\\nolimits ^6(\\mathop{\\mathrm{Sym}}\\nolimits ^bV)`$ embeds equivariantly in $`\\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^6V)`$ for every b ≥ 6 and finite-dimensional complex V.", "coveredBy": [ "FoulkesHowe" ], "note": "Covered only for b ≥ 30 = 6·5 and only implicitly: FoulkesHowe gives surjectivity of the canonical map Sym^b(Sym^6 V) → Sym^6(Sym^b V) for b ≥ 30; the equivariant injection Sym^6(Sym^b V) ↪ Sym^b(Sym^6 V) follows by GL(V)-equivariance of the canonical map and complete reducibility, neither of which is stated. The range 6 ≤ b < 30 is declared out." }, { "clause": "More generally, the canonical multiplication map $`\\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^aV)\\to\\mathop{\\mathrm{Sym}}\\nolimits ^a(\\mathop{\\mathrm{Sym}}\\nolimits ^bV)`$ is surjective for a ≥ 2 and $`b\\ge a(a-1)`$, giving dimension-independent quadratic stabilization.", "coveredBy": [ "FoulkesHowe" ], "note": "∀ finite-dimensional complex V, a ≥ 2, b ≥ a(a−1): the unique linear map satisfying the averaged Foulkes–Howe formula on monomials exists and is surjective." } ], "gap": "The sixth Foulkes case for 6 ≤ b < 30 is not stated (only b ≥ 30 follows), and the equivariant embedding itself is not stated even for b ≥ 30.", "scopeNoteOnGap": "The sixth-power specialization is covered for $b\\ge30$; the companion's full range $b\\ge6$ is not included.", "note": null, "firstSuspicious": "Like 132, the scope note credits 'the formalization' with results (the a = 1 case, vanishing on products, the equivariant embedding) that are not in the Comparator statement. The headline's own clause 1 is still declared narrower (b ≥ 30), so the family word does not depend on this.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "FoulkesHowe", "flag": "The scope note says the formalization 'also covers the bijective a=1 case, the vanishing-on-products consequence, and the corresponding equivariant embedding in the reverse direction' (lean/docs/210.md:9). None of these is in the Comparator statement; they appear in the formal library (e.g. OAI/RepresentationTheory/FoulkesHowe/PlethysmDual.lean:88 exists_embedding_of_dual_surjection), outside Comparator's check." } ], "flagReadings": null, "readings": [ "first/batch3.json" ] }, { "family": "211", "title": "The geometric phase diagram, diffusion, and spectra of random planar maps", "headline": "Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for $`0\\lt q\\le4`$ and to the Brownian continuum random tree for q > 4, establishing the surface-to-tree geometric transition. For FK–Ising and spanning-tree-weighted maps, stationary random walks converge to Liouville Brownian motion on the limiting sphere. The FK–Ising spectral result also gives convergence of eigenvalues and heat traces, using the stated Brownian/LQG inputs.", "challenges": [ "FKCRT" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for $`0\\lt q\\le4`$", "status": "unlinked", "coveredBy": [], "manuscript": "Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026 ; The-critical-Liouville-quantum-sphere-and-geometric-limits-of-FK-maps-at-q-equals-4-September-24-2026 ; Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026", "evidence": [ "preprints/Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026/build/main.tex:9-12: For every fixed $0 4", "status": "stated", "coveredBy": [ "FKCRT" ], "encoding": "rooted genus-0 rotation systems with n+1 edges up to relabelling; FK weight q^{k(A)+(|A|−|V|)/2}; degree measure; GHP convergence in law as convergence of bounded GHP-continuous tests; the limit tree coded by the 3D Bessel bridge (standard excursion), constant c absorbing the normalization", "evidence": [ "lean/ComparatorChallenges/FKCRT.lean:194-204: def MainStatement : Prop := ⏎ ∀ q : ℝ, 4 < q → ∃ c : ℝ, 0 < c ∧ ⏎ ∀ (Ω : Type) (mΩ : MeasurableSpace Ω) (P : @Measure Ω mΩ), ⏎ @IsProbabilityMeasure Ω mΩ P → ⏎ ∀ B : Fin 3 → ℝ≥0 → Ω → ℝ, ⏎ (∀ i, @IsBrownianReal Ω mΩ (B i) P) → ⏎ @iIndepFun Ω (Fin 3) mΩ (fun _ => ℝ≥0 → ℝ) (fun _ => inferInstance) ⏎ (fun i ω t => B i t ω) P → ⏎ ∀ F : MetricMeasureData → ℝ, IsGHPTest F → ⏎ Tendsto (fun n : ℕ => finiteExpectation q (c / Real.sqrt (n + 1)) n F) ⏎ atTop (𝓝 (@brownianExpectation Ω mΩ P B F))" ] }, { "clause": "establishing the surface-to-tree geometric transition.", "status": "unlinked", "coveredBy": [], "manuscript": "Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026 ; The-critical-Liouville-quantum-sphere-and-geometric-limits-of-FK-maps-at-q-equals-4-September-24-2026 ; Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026", "evidence": [ "preprints/Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four-September-24-2026/build/sections/history.tex:26-31: The contrast with the other FK regimes is geometric. Companion results ⏎ for $04 CRT manuscript and describes only it (docs/211.md:9: \"For every fixed $q>4$, the formalization proves the Brownian continuum-random-tree limit for critical finite Fortuin–Kasteleyn planar maps.\"); nothing is said about the omitted clauses", "note": "Only the q > 4 CRT limit is stated, and it is the only paper the note links (one of seven). The transition clause needs the q ≤ 4 side, which the linked CRT paper itself calls context, not input (history.tex:26-31). Every missing clause is from an unlinked paper, so UNLINKED; differs from the first reader only by amendment 3.", "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "FKCRT", "flag": "JUNK? cleared: brownianExpectation is a Bochner integral that would be 0 for a non-integrable test, but a junk 0 could only make the convergence claim false (constant tests force the limit), not trivial" }, { "challenge": "FKCRT", "flag": "VACUITY? cleared: three independent Brownian motions exist, so the ∀-over-realizations quantifier is not vacuous" } ], "flagReadings": null, "readings": [ "first/batch8.json", "second/second6.json" ] }, { "family": "212", "title": "Planar first-passage geometry and the absence of bigeodesics", "headline": "Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with C1 boundary. Differentiability also holds for every Gamma law with positive shape and rate.", "challenges": [ "PlanarFirstPassage", "GammaPassage" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment.", "status": "unlinked", "coveredBy": [], "manuscript": "No-bigeodesics-in-planar-first-passage-percolation-September-24-2026", "evidence": [ "CONTENTS.md:4891: [No bigeodesics in planar first-passage percolation]", "CONTENTS.md:4893: We prove that planar first-passage percolation with independent identically distributed nonnegative nonatomic edge weights has almost surely no doubly infinite geodesic, provided the minimum of four independent weights has finite second moment.", "lean/docs/212.md:5: - [Strict convexity and differentiability of the planar exponential first-passage limit shape](../../preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026/main.pdf)", "lean/docs/212.md:1: # Planar first-passage geometry and the absence of bigeodesics" ], "note": "No challenge mentions geodesics; the note's title names bigeodesics, but a title alone does not declare, and the only linked paper is the strict-convexity/differentiability one." }, { "clause": "For exponential weights, the limit shape is strictly convex", "status": "declared", "coveredBy": [], "manuscript": "Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026", "evidence": [ "lean/docs/212.md:11: The paper also claims strict convexity for the exponential limit shape. Strict convexity is outside these selected differentiability statements." ] }, { "clause": "with C1 boundary.", "status": "stated", "coveredBy": [ "PlanarFirstPassage" ], "evidence": [ "lean/ComparatorChallenges/PlanarFirstPassage.lean:62: marginal : ∀ e, Measure.map (τ e) P = expMeasure 1", "lean/ComparatorChallenges/PlanarFirstPassage.lean:94: DifferentiableAwayFromOrigin μ ∧ UniqueNormalizedSupports μ ∧ C1UnitSphere μ ∧", "lean/ComparatorChallenges/PlanarFirstPassage.lean:97: (∀ v ∈ frontier {z | μ z ≤ 1}, HasC1CurveChart (frontier {z | μ z ≤ 1}) v) := by" ], "note": "Encoding: rate-1 exponential weights (other rates by scaling, and GammaPassage with shape 1 covers every rate); the limit shape is the unit ball of the a.s. radial time constant μ." }, { "clause": "Differentiability also holds for every Gamma law with positive shape and rate.", "status": "stated", "coveredBy": [ "GammaPassage" ], "evidence": [ "lean/ComparatorChallenges/GammaPassage.lean:97: ∀ (shape rate : ℝ), 0 < shape → 0 < rate →", "lean/ComparatorChallenges/GammaPassage.lean:103: DifferentiableAwayFromOrigin μ ∧" ] } ], "gap": "The no-bigeodesics theorem (the headline's first sentence) is not stated; strict convexity of the exponential limit shape is not stated (declared).", "scopeNoteOnGap": "No-bigeodesics: silent — the bigeodesics paper is not linked; only the title mentions it: lean/docs/212.md:1: the absence of bigeodesics. Strict convexity: lean/docs/212.md:11: Strict convexity is outside these selected differentiability statements.", "note": "C¹ boundary (exponential) and Gamma differentiability are stated; strict convexity is declared (docs/212.md:11); the no-bigeodesics theorem comes from a manuscript docs/212.md does not link: NARROWER — UNLINKED. The first reader recorded the same three facts and called the bigeodesics gap UNDECLARED before amendment 3; the difference is the amendment.", "firstSuspicious": "Pattern seen across this batch: the CONTENTS.md headline was widened to manuscripts the scope note does not link, and the scope note never says those manuscripts are unformalized. Strictly that is silence, hence UNDECLARED; a second reader should weigh whether the scope note's paper list counts as a declaration. Here: the September 24 no-bigeodesics paper.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json", "second/second7.json" ] }, { "family": "213", "title": "Critical percolation on every quasi-transitive graph", "headline": "Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with $`p_c\\lt 1`$: at the critical probability, there is almost surely no infinite cluster. The family also establishes this conclusion for both nearest-neighbor bond and site percolation on ℤ3.", "challenges": [ "CriticalZ3", "CriticalPercolation" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with $`p_c\\lt 1`$: at the critical probability, there is almost surely no infinite cluster.", "coveredBy": [ "CriticalPercolation" ], "note": "For every bond graph (multi-bonds and loops allowed) on an infinite vertex type that is connected, locally finite (finitely many bonds at each vertex), quasi-transitive (finitely many Aut-orbits) with p_c < 1: P_{p_c}(some cluster is infinite) = 0." }, { "clause": "The family also establishes this conclusion for both nearest-neighbor bond and site percolation on ℤ3.", "coveredBy": [ "CriticalZ3" ], "note": "At the bond and site critical parameters of nearest-neighbour ℤ³, a.s. every cluster is finite." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "CriticalZ3", "flag": "JUNK? (checked, benign): bondCritical/siteCritical are sInf over ℝ (lines 50-54); sInf ∅ = 0 would make the claim trivial at p = 0, but the set is nonempty (p = 1 percolates surely), so the sInf is the genuine p_c." } ], "flagReadings": [ { "challenge": "CriticalZ3", "flag": "JUNK? (checked, benign): bondCritical/siteCritical are sInf over ℝ (lines 50-54); sInf ∅ = 0 would make the claim trivial at p = 0, but the set is nonempty (p = 1 percolates surely), so the sInf is the genuine p_c.", "decision": "cleared", "why": "Mathlib defines bernoulliMeasure x y p = p•δx + (1−p)•δy, so each bond or site is open with probability p. Both sets {p ∈ [0,1] | 0 < θ(p)} are bounded below by 0. They are nonempty because at p = 1 the law is the Dirac mass at the all-open configuration, whose cluster at 0 is all of ℤ³ (and for sites, ω 0 = true). bondCritical and siteCritical are therefore the genuine infima, i.e. p_c by monotonicity. The junk sInf ∅ = 0 would have moved the claim to p = 0, where it is trivially true, but that value cannot be reached.", "evidence": [ "lean/ComparatorChallenges/CriticalZ3.lean:45: Measure.infinitePi (fun _ : Bond => bernoulliMeasure true false (parameter p))", "lean/ComparatorChallenges/CriticalZ3.lean:48: Measure.infinitePi (fun _ : Vertex => bernoulliMeasure true false (parameter p))", "lean/ComparatorChallenges/CriticalZ3.lean:50: noncomputable def bondCritical : ℝ :=", "lean/ComparatorChallenges/CriticalZ3.lean:51: sInf {p : ℝ | p ∈ Set.Icc (0 : ℝ) 1 ∧ 0 < bondLaw p (bondInfiniteAt 0)}", "lean/ComparatorChallenges/CriticalZ3.lean:53: noncomputable def siteCritical : ℝ :=", "lean/ComparatorChallenges/CriticalZ3.lean:54: sInf {p : ℝ | p ∈ Set.Icc (0 : ℝ) 1 ∧ 0 < siteLaw p (siteInfiniteAt 0)}", "lean/ComparatorChallenges/CriticalZ3.lean:57: (∀ᵐ ω ∂bondLaw bondCritical, ∀ x : Vertex, (bondCluster ω x).Finite) ∧", "lean/ComparatorChallenges/CriticalZ3.lean:58: (∀ᵐ ω ∂siteLaw siteCritical, ∀ x : Vertex, (siteCluster ω x).Finite) := by" ] } ], "readings": [ "first/batch3.json", "flags/flags2.json" ] }, { "family": "214", "title": "The Benjamini–Schramm nonuniqueness conjecture", "headline": "Proves $`p_c\\lt p_u`$ for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition.", "challenges": [ "BenjaminiSchramm", "CayleyPercolation" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves $`p_c\\lt p_u`$ for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture.", "coveredBy": [ "BenjaminiSchramm", "CayleyPercolation" ], "note": "full_main: for infinite V, connected, locally finite, quasi-transitive, vertex-isoperimetric constant > 0: pc < p_{2→2} ≤ pu." }, { "clause": "Thus there is a nonempty range of probabilities with infinitely many infinite clusters.", "coveredBy": [ "BenjaminiSchramm", "CayleyPercolation" ], "note": "A coupled interval [p₁, p₂] with pc < p₁ < p₂ < 1 on which, a.s. for all p in it, there are infinitely many infinite clusters; for Cayley graphs, a.s. infinitely many infinite clusters at every p ∈ (pc, pu)." }, { "clause": "A stronger operator bound also establishes the critical triangle condition.", "coveredBy": [ "BenjaminiSchramm", "CayleyPercolation" ], "note": "‖T_{pc}‖_{2→2} < ∞ (and continuity from below), and critical_laws: triangle diagram ∇_{pc}(x) ≤ ‖T_{pc}‖³ < ∞, plus mean-field critical exponents." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "BenjaminiSchramm", "flag": "META: the file contains 'run_elab Lean.modifyEnv fun environment => Lean.Meta.auxLemmasExt.setState environment {}' five times (lines 92, 108, 120, 180, 195). It resets the auxiliary-lemma cache and does not appear to change any stated proposition, but it is environment-modifying code inside a statement file; a second reader should confirm it is benign." }, { "challenge": "BenjaminiSchramm", "flag": "JUNK? checked, none: the susceptibility tsum would be 0 if not summable, but the stated lower bound c/(pc − p) > 0 excludes that; the triangle diagram is an ℝ≥0∞ tsum; graph distance is junk 0 only on disconnected graphs, which are excluded." } ], "flagReadings": [ { "challenge": "BenjaminiSchramm", "flag": "JUNK? checked, none: the susceptibility tsum would be 0 if not summable, but the stated lower bound c/(pc − p) > 0 excludes that; the triangle diagram is an ℝ≥0∞ tsum; graph distance is junk 0 only on disconnected graphs, which are excluded.", "decision": "cleared", "why": "Each junk route checked. (1) susceptibility is a real tsum and would be 0 if not summable; SusceptibilityLaw asserts c/(pc−p) ≤ χ with c > 0, so the junk can only falsify. It is also not reached: χ(p) < ∞ for p < pc on quasi-transitive graphs (sharpness of the phase transition). The window p < pc is nonempty, because quasi-transitive plus locally finite gives bounded degree and hence pc ≥ 1/(Δ−1) > 0, so the law is not vacuous. (2) triangleDiagram is an ℝ≥0∞ tsum, so no summability junk. operatorNorm = sInf ∅ = ⊤ is the honest value for an unbounded kernel, not junk. ⊤ would make `≤ ‖T‖³` trivial, but `triangleDiagram … < ⊤` is a separate conjunct, and full_main asserts operatorNorm (pc) < ⊤. (3) graphDistance = SimpleGraph.dist is 0 only between unreachable vertices. hc : Connected G gives reachability in openSimpleGraph G univ (dropping loops does not affect connectivity). The intrinsic distance is taken only for y in the open cluster of x, hence reachable. (4) The hypothesis hV is a Real sInf over a nonempty set bounded below by 0, and vertexBoundary is finite under LocallyFinite, so ncard is not junk 0. `0 < hV G` is genuine nonamenability, not an unsatisfiable hypothesis. Lane K: not run yet.", "evidence": [ "ComparatorChallenges/BenjaminiSchramm.lean:16: def LocallyFinite (G : BondGraph V E) : Prop := ∀ x, {e | x ∈ G e}.Finite", "ComparatorChallenges/BenjaminiSchramm.lean:31: noncomputable def hV (G : BondGraph V E) : ℝ :=", "ComparatorChallenges/BenjaminiSchramm.lean:32: sInf {r | ∃ A : Finset V, A.Nonempty ∧", "ComparatorChallenges/BenjaminiSchramm.lean:33: r = (vertexBoundary G A).ncard / (A.card : ℝ)}", "ComparatorChallenges/BenjaminiSchramm.lean:67: noncomputable def operatorNorm (G : BondGraph V E) (p : unitInterval) : ℝ≥0∞ := by", "ComparatorChallenges/BenjaminiSchramm.lean:68: classical", "ComparatorChallenges/BenjaminiSchramm.lean:69: exact sInf {r | ∃ A : Hilbert V →L[ℂ] Hilbert V,", "ComparatorChallenges/BenjaminiSchramm.lean:70: HasKernel G p A ∧ r = ENNReal.ofReal ‖A‖}", "ComparatorChallenges/BenjaminiSchramm.lean:115: noncomputable def susceptibility (G : BondGraph V E) (p : unitInterval) (x : V) : ℝ :=", "ComparatorChallenges/BenjaminiSchramm.lean:116: ∑' y, twoPoint G p x y", "ComparatorChallenges/BenjaminiSchramm.lean:127: noncomputable def triangleDiagram (G : BondGraph V E) (p : unitInterval) (x : V) : ℝ≥0∞ :=", "ComparatorChallenges/BenjaminiSchramm.lean:128: ∑' ab : V × V, ENNReal.ofReal", "ComparatorChallenges/BenjaminiSchramm.lean:129: (twoPoint G p x ab.1 * twoPoint G p ab.1 ab.2 * twoPoint G p ab.2 x)", "ComparatorChallenges/BenjaminiSchramm.lean:130: ", "ComparatorChallenges/BenjaminiSchramm.lean:131: noncomputable def graphDistance (G : BondGraph V E) (x y : V) : ℕ :=", "ComparatorChallenges/BenjaminiSchramm.lean:132: (openSimpleGraph G Set.univ).dist x y", "ComparatorChallenges/BenjaminiSchramm.lean:163: def SusceptibilityLaw (G : BondGraph V E) : Prop :=", "ComparatorChallenges/BenjaminiSchramm.lean:164: ∀ x, ∃ c C δ : ℝ, 0 < c ∧ 0 < C ∧ 0 < δ ∧", "ComparatorChallenges/BenjaminiSchramm.lean:165: ∀ p : unitInterval, p < pc G → (pc G : ℝ) - p < δ →", "ComparatorChallenges/BenjaminiSchramm.lean:166: c / ((pc G : ℝ) - p) ≤ susceptibility G p x ∧", "ComparatorChallenges/BenjaminiSchramm.lean:167: susceptibility G p x ≤ C / ((pc G : ℝ) - p)", "ComparatorChallenges/BenjaminiSchramm.lean:186: def CriticalLawsConclusion (G : BondGraph V E) : Prop :=", "ComparatorChallenges/BenjaminiSchramm.lean:187: (∀ x, triangleDiagram G (pc G) x ≤ operatorNorm G (pc G) ^ 3 ∧", "ComparatorChallenges/BenjaminiSchramm.lean:188: triangleDiagram G (pc G) x < ⊤) ∧", "ComparatorChallenges/BenjaminiSchramm.lean:217: theorem full_main {V : Type u} {E : Type v} [Infinite V]", "ComparatorChallenges/BenjaminiSchramm.lean:218: (G : BondGraph V E) (hc : Connected G) (hl : LocallyFinite G)", "ComparatorChallenges/BenjaminiSchramm.lean:219: (hq : QuasiTransitive G) (hh : 0 < hV G) : MainConclusion G := by" ] } ], "readings": [ "first/batch5.json", "flags/flags3.json" ] }, { "family": "215", "title": "Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics", "headline": "Constructs the canonical continuum limit of the two-dimensional nearest-neighbor $`O(3)`$ model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice $`O(4)`$ model, determines the exact leading asymptotic of the full transfer gap, $`m_{\\mathrm{lat}}(\\beta)\\sim32e^{\\pi/4-1/2}\\sqrt\\beta\\,e^{-\\pi\\beta}`$. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor $`O(n)`$ models with n ≥ 3 at every positive temperature.", "challenges": [ "ClassicalON" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Constructs the canonical continuum limit of the two-dimensional nearest-neighbor $`O(3)`$ model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap.", "status": "unlinked", "coveredBy": [], "manuscript": "The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026", "evidence": [ "corpus/openai-math/release.json:9555: \"dir\": \"The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026\"", "corpus/openai-math/release.json:9557: We construct a canonical interacting massive continuum limit of the two-dimensional nearest-neighbor $`O(3)`$ model", "corpus/openai-math/release.json:9557: Their reconstructed theory has a unique vacuum, a nonzero vacuum complement, and a positive Hamiltonian gap on that entire complement.", "lean/docs/215.md:3: The following describes the scope of the Lean formalization related to the following accompanying paper(s):", "lean/docs/215.md:5: - [Exponential decay in two-dimensional classical $O(n)$ models](../../preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026/paper.pdf)", "lean/docs/215.md:9: The infinite-volume limit and the separate $O(4)$ spectral-gap claim are not included." ] }, { "clause": "For the square-lattice $`O(4)`$ model, determines the exact leading asymptotic of the full transfer gap, $`m_{\\mathrm{lat}}(\\beta)\\sim32e^{\\pi/4-1/2}\\sqrt\\beta\\,e^{-\\pi\\beta}`$.", "status": "declared", "coveredBy": [], "manuscript": "Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026", "evidence": [ "lean/docs/215.md:9: The infinite-volume limit and the separate $O(4)$ spectral-gap claim are not included.", "corpus/openai-math/release.json:9577: \"dir\": \"Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026\"", "corpus/openai-math/release.json:9579: Here the mass is the gap of the full Osterwalder–Schrader transfer operator" ] }, { "clause": "The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor $`O(n)`$ models with n ≥ 3 at every positive temperature.", "status": "stated", "coveredBy": [ "ClassicalON" ], "evidence": [ "lean/ComparatorChallenges/ClassicalON.lean:46: def ExponentialDecay : Prop :=", "lean/ComparatorChallenges/ClassicalON.lean:47: ∀ (n : ℕ), 3 ≤ n → ∀ (β : ℝ), 0 < β →", "lean/ComparatorChallenges/ClassicalON.lean:48: ∃ A m : ℝ, 0 < m ∧ ∀ (G : LatticeGraph) (b : G.edges → ℝ),", "lean/ComparatorChallenges/ClassicalON.lean:49: (∀ e, 0 ≤ b e ∧ b e ≤ β) → ∀ x y : G.vertices,", "lean/ComparatorChallenges/ClassicalON.lean:50: 0 ≤ correlation n G b x y ∧", "lean/ComparatorChallenges/ClassicalON.lean:51: correlation n G b x y ≤ A * Real.exp (-m * siteDistance x.val y.val)", "encoding: uniform over finite square-lattice subgraphs with free boundary and edge strengths in [0, β], as in the manuscript; the infinite-volume limit is declared out (215.md:9)" ] } ], "gap": "the O(3) canonical continuum limit (undeclared) and the O(4) exact mass asymptotic (declared)", "scopeNoteOnGap": "The infinite-volume limit and the separate $O(4)$ spectral-gap claim are not included. (docs/215.md:9) — declares the O(4) clause; silent on the O(3) continuum limit, whose two manuscripts are not linked", "note": "The O(n) decay clause is stated (finite-volume, uniform). The O(4) mass asymptotic is declared out by name (215.md:9) although its manuscript is unlinked; the O(3) continuum-limit clause is neither stated nor named and comes from unlinked manuscripts, hence UNLINKED. I differ from the first reader only through amendment 3.", "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "ClassicalON", "flag": "JUNK? cleared: the integrands are bounded and continuous on a product of probability spaces, so partition > 0 and no Bochner junk" } ], "flagReadings": null, "readings": [ "first/batch8.json", "second/second1.json" ] }, { "family": "218", "title": "Conformal universality for weakly interacting and random-bond Ising models", "headline": "Weak finite-range square-symmetric even multispin perturbations of the square-lattice Ising model preserve critical bulk spin and energy limits; weak square-symmetric contour interactions also yield chordal SLE3 interface limits. With sufficiently weak iid bond disorder of any fixed bounded nondegenerate mean-zero law, critical spin interfaces converge to the same law in probability over environments.", "challenges": [ "BufferedIsing" ], "word": "DIFFERENT", "firstWord": "DIFFERENT", "secondWord": "DIFFERENT", "disagreement": null, "clauses": [ { "clause": "Weak finite-range square-symmetric even multispin perturbations of the square-lattice Ising model preserve critical bulk spin and energy limits", "status": "unlinked", "coveredBy": [], "manuscript": "Conformal-universality-of-bulk-Ising-correlations-under-weak-interactions-September-23-2026", "evidence": [ "lean/docs/218.md:5: - [Buffered comparison and stopping-band resolution in critical Ising](../../preprints/Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026/paper.pdf)", "CONTENTS.md:5136: We prove conformal universality of mixed bulk spin and energy correlations for the square-lattice Ising model with sufficiently small, square-symmetric, finite-range even multispin perturbations of either sign." ], "note": "No scaling limit, correlation function, lattice or multispin perturbation appears in BufferedIsing." }, { "clause": "weak square-symmetric contour interactions also yield chordal SLE3 interface limits.", "status": "unlinked", "coveredBy": [], "manuscript": "SLE3-universality-for-weak-finite-range-Ising-interactions-September-23-2026", "evidence": [ "lean/docs/218.md:5: - [Buffered comparison and stopping-band resolution in critical Ising]", "CONTENTS.md:5145: We prove that every sufficiently small, square-symmetric finite-range perturbation of the planar Ising contour energy has a domain-independent inverse temperature at which its spin interface converges to chordal SLE3." ], "note": "No SLE, interface or curve law appears." }, { "clause": "With sufficiently weak iid bond disorder of any fixed bounded nondegenerate mean-zero law, critical spin interfaces converge to the same law in probability over environments.", "status": "unlinked", "coveredBy": [], "manuscript": "Quenched-SLE3-limits-for-general-weak-random-bond-Ising-models-October-5-2026", "evidence": [ "lean/docs/218.md:5: - [Buffered comparison and stopping-band resolution in critical Ising]", "CONTENTS.md:5109: where the ξe have any fixed bounded, nondegenerate, mean-zero iid law, we prove quenched chordal SLE3 convergence for sufficiently small ε > 0." ], "note": "No random environment or quenched convergence appears. The symmetric two-valued special case is the also-unlinked Quenched-SLE3-Universality-for-the-Weak-Random-Bond-Ising-Model-October-5-2026." } ], "gap": "None of the headline's three claims (bulk conformal universality, SLE₃ interface limits for contour perturbations, quenched SLE₃ for weak random bonds) is stated. The family's only challenge is a finite-graph Ising likelihood-ratio inequality — a proof ingredient from one of the family's six manuscripts ('Buffered comparison and stopping-band resolution in critical Ising'). It neither implies the headline nor is implied by it, and it is not among the supporting results declared in ComparatorChallenges/README.md.", "scopeNoteOnGap": "silent on the headline's claims — docs/218.md lists only the buffered-comparison manuscript (line 5) and describes only the finite-graph comparison (line 9: 'The formalized result compares conditional Ising probabilities on finite graphs when two boundary mixtures differ but a common set of spins is pinned.'); its only stated exclusion is 'The paper's finite stopping-band approximation theorem is outside this selected statement.' It never says the universality / SLE₃ / random-bond results are not formalized.", "note": "No headline clause is stated. The one challenge is a finite-graph Ising likelihood-ratio inequality (exp(±4 artanh q₀), q₀ a zero-field FK connection probability) from the linked buffered-comparison paper: it neither implies nor is implied by any headline clause, so the family is DIFFERENT. In fairness, every headline clause comes from a manuscript the note does not link, and docs/218.md:9 describes exactly what it formalizes (it also declares the stopping-band theorem out). The DIFFERENT comes from the CONTENTS headline, not from an inaccurate note. Agrees with the first reader. Junk checks pass: all denominators are positive, and q₀ < 1 because the empty FK configuration never joins I to J.", "firstSuspicious": "Six manuscripts, one challenge, and that challenge is a finite-graph lemma: whether to call it DIFFERENT (my reading: it neither implies nor is implied by any headline clause) or NARROWER — UNDECLARED (it is 'less' than the headline in scope) depends on how the page reads 'states less'; logically it is not a weakening of any headline clause.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json", "second/second2.json" ] }, { "family": "220", "title": "Directional zero–one laws beyond iid environments and iid ballisticity", "headline": "On ℤd, d ≥ 3, directional escape has probability zero or one for iid strictly elliptic nearest-neighbor environments, and for stationary ergodic finite-range-dependent environments under uniform ellipticity. In iid uniformly elliptic environments with d ≥ 2, almost-sure directional transience implies a deterministic limiting velocity with positive projection in that direction, resolving the ballisticity conjecture.", "challenges": [ "DirectionalWalk", "DirectionalBallisticity", "VelocityHemisphere" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "On ℤd, d ≥ 3, directional escape has probability zero or one for iid strictly elliptic nearest-neighbor environments", "status": "stated", "coveredBy": [ "DirectionalWalk" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/DirectionalWalk.lean:88: annealed μ 0 (escape ℓ) = 0 ∨ annealed μ 0 (escape ℓ) = 1 := by sorry" ] }, { "clause": "and for stationary ergodic finite-range-dependent environments under uniform ellipticity.", "status": "unlinked", "coveredBy": [], "manuscript": "A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026", "evidence": [ "lean/ComparatorChallenges/DirectionalWalk.lean:32: Measure.infinitePi (fun _ => μ)", "lean/ComparatorChallenges/DirectionalBallisticity.lean:22: Measure (Environment d) := Measure.infinitePi (fun _ => ν)", "lean/ComparatorChallenges/VelocityHemisphere.lean:22: Measure (Environment d) := Measure.infinitePi (fun _ => ν)", "preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/main.tex:38: random walks in stationary, ergodic, finite-range-dependent environments on", "lean/docs/220.md:5: - [A directional zero–one law under strict ellipticity]", "lean/docs/220.md:6: - [Directional transience implies ballisticity]", "lean/docs/220.md:1: # Directional zero–one laws beyond iid environments and iid ballisticity" ] }, { "clause": "In iid uniformly elliptic environments with d ≥ 2, almost-sure directional transience implies a deterministic limiting velocity with positive projection in that direction, resolving the ballisticity conjecture.", "status": "stated", "coveredBy": [ "DirectionalBallisticity", "VelocityHemisphere" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/DirectionalBallisticity.lean:90: ∀ (d : ℕ), 2 ≤ d → ∀ (ν : Measure (Row d)), IsProbabilityMeasure ν →", "lean/ComparatorChallenges/DirectionalBallisticity.lean:93: ∃ v : Vector d, 0 < dot v ℓ ∧ HasVelocity ν v" ] } ], "gap": "The directional zero–one law for stationary ergodic finite-range-dependent uniformly elliptic environments is claimed but not stated by any family-220 challenge.", "scopeNoteOnGap": "silent", "note": "All three statements build the environment as an i.i.d. product (Measure.infinitePi of one row law), so the finite-range-dependent clause has no counterpart; it is the October 5 manuscript, which the note does not link (the note's heading says 'beyond iid environments', but a heading does not declare, and its body describes only i.i.d. laws). Every clause from the two linked papers is stated (annealed law from the origin, environment sampled once and kept; unit ℓ in the ballisticity statement). NARROWER — UNLINKED; I differ from the first reader by the amendment-3 word only.", "firstSuspicious": "The headline was extended with the October 5 finite-range-dependent paper; lean/docs/220.md still links only the two September papers (lines 5-6) and describes only i.i.d. environments.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "DirectionalWalk", "flag": "NARROWER — UNDECLARED (family level): environmentLaw is i.i.d. (line 30-32); the finite-range-dependent clause has no counterpart and lean/docs/220.md is silent on it." } ], "flagReadings": null, "readings": [ "first/batch3.json", "second/second3.json" ] }, { "family": "221", "title": "The Mézard–Parisi formula for diluted spin glasses", "headline": "Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability.", "challenges": [ "DilutedSpin" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability", "coveredBy": [ "DilutedSpin" ], "note": "Admissible: even p >= 2, alpha > 0, first moments of ||theta||_inf and |h|, the factorization exp theta = a(1 + b prod f_l) with |b prod f| < 1, f_l iid and independent of b, all moments of b integrable, and E(-b)^n >= 0." }, { "clause": "The limiting free energy equals the infimum over finite-depth hierarchical trial laws", "coveredBy": [ "DilutedSpin" ], "note": "pressure M N -> inf over r of sInf {functional M r zeta m : zeta in H_{r+1}, 0 < m_1 < ... < m_r < 1}; existence of the limit is part of the assertion." }, { "clause": "This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability", "coveredBy": [ "DilutedSpin" ], "note": "Covered by instantiation, not stated separately. I checked that Viana-Bray satisfies Admissible (p = 2, exp(beta J s s') = cosh(beta J)(1 + tanh(beta J) s s'), symmetric J); I did not check the other two families. The Lean theorem has no hidden restriction that would exclude them." } ], "gap": null, "scopeNoteOnGap": "no gap", "note": null, "firstSuspicious": "Measurability of the hierarchical trial integrands (see the JUNK? flag) is the only point I could not close by reading. Confidence is medium for that reason; the clause coverage itself is high.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [ { "challenge": "DilutedSpin", "flag": "JUNK? (low): trialLog and functional are Bochner integrals over the nested weak-topology hierarchy. An integrand that is not AE-strongly measurable integrates to 0 and could pull phi r below the true infimum. That would make the Lean equality FALSE, not weaker. I argued measurability by continuity of eta -> logMean on the Polish hierarchy spaces (induction on r) but did not verify it line by line. Lean: ComparatorChallenges/DilutedSpin.lean:111-113, 119-120." } ], "flagReadings": [ { "challenge": "DilutedSpin", "flag": "JUNK? (low): trialLog and functional are Bochner integrals over the nested weak-topology hierarchy. An integrand that is not AE-strongly measurable integrates to 0 and could pull phi r below the true infimum. That would make the Lean equality FALSE, not weaker. I argued measurability by continuity of eta -> logMean on the Polish hierarchy spaces (induction on r) but did not verify it line by line. Lean: ComparatorChallenges/DilutedSpin.lean:111-113, 119-120.", "decision": "cleared", "why": "No Bochner junk can occur, and the first reader's open measurability point closes. (1) Every integrand in the functional is bounded, independently of the trial law ζ. q(x,·) is a probability vector (e^{±x}/(2cosh x)), so edge θ x and the sum inside message are convex combinations of e^{θ(s)}. That gives |log edge θ x| ≤ ‖θ‖ and |message| ≤ ‖θ‖. siteLog is the log of an average over ε, so |siteLog θ h x| ≤ |h| + Σ_j ‖θ_j.1‖. logMean preserves a bound C at every level (a log-power-mean with m 0 > 0 by Exponents), so heavy-tailed ζ cannot make anything non-integrable. The outer integrals are dominated by |h| + Σ_j‖θ_j‖ and ‖θ‖, which are integrable by interaction_integrable and field_integrable against a Poisson number of terms (finite mean). (2) Measurability: siteLog and log∘edge are continuous in the messages. Each Hierarchy r is Polish (ℝ, then P(Polish) with the weak topology; the σ-algebra is Borel either way). For bounded continuous F, η ↦ ∫F d(⊗_i η_i) is weakly continuous on finite products of separable metric spaces, so by induction η ↦ logMean r g m η is continuous, hence Borel. (3) pressure is genuine: |logPartition| ≤ N log 2 + Σ_j‖θ_j.1‖ + Σ_i|h_i|, continuous, integrable. So phi r is the infimum of a genuine, finite functional. Because the left side is genuine, any residual junk on the right could only make the equality false. Lane K: CERTIFIED.", "evidence": [ "ComparatorChallenges/DilutedSpin.lean:33-35: interaction_integrable : Integrable (fun z : InteractionSample p => ‖z.1‖) ⏎ M.disorder.toMeasure ⏎ field_integrable : Integrable (fun h : ℝ => |h|) M.field.toMeasure", "ComparatorChallenges/DilutedSpin.lean:58-72: noncomputable def logPartition {p N k : ℕ} (theta : Fin k → InteractionSample p) ⏎ (h : Fin N → ℝ) (indices : Fin k → Fin p → Fin N) : ℝ := ⏎ Real.log (∑ σ : Fin N → Spin, Real.exp (logWeight theta h indices σ)) ⏎ ⏎ /-- F_N, using an explicit uniform average over the independent index labels. ⏎ The value at N=0 is irrelevant to the limit; valid system sizes are N>=1. -/ ⏎ noncomputable def pressure {p : ℕ} (M : Model p) (N : ℕ) : ℝ := ⏎ (∫ k : ℕ, ⏎ ∫ theta : Fin k → InteractionSample p, ⏎ ∫ h : Fin N → ℝ, ⏎ ((∑ indices : Fin k → Fin p → Fin N, logPartition theta h indices) / ⏎ (Fintype.card (Fin k → Fin p → Fin N) : ℝ)) ⏎ ∂Measure.pi (fun _ : Fin N => M.field.toMeasure) ⏎ ∂Measure.pi (fun _ : Fin k => M.disorder.toMeasure) ⏎ ∂poissonMeasure (M.alpha * N)) / N", "ComparatorChallenges/DilutedSpin.lean:74-78: noncomputable def q (x : ℝ) (s : Spin) : ℝ := ⏎ Real.exp (x * spin s) / (2 * Real.cosh x) ⏎ ⏎ noncomputable def edge {p : ℕ} (theta : Interaction p) (x : Fin p → ℝ) : ℝ := ⏎ ∑ s : Fin p → Spin, Real.exp (theta s) * ∏ l, q (x l) (s l)", "ComparatorChallenges/DilutedSpin.lean:84-92: noncomputable def message {p : ℕ} (theta : Interaction p) ⏎ (x : Fin (p - 1) → ℝ) (ε : Spin) : ℝ := ⏎ Real.log (∑ s : Fin (p - 1) → Spin, ⏎ Real.exp (theta (appendSpin s ε)) * ∏ l, q (x l) (s l)) ⏎ ⏎ noncomputable def siteLog {p k : ℕ} (theta : Fin k → InteractionSample p) ⏎ (h : ℝ) (x : (Fin k × Fin (p - 1)) → ℝ) : ℝ := ⏎ Real.log ((∑ ε : Spin, Real.exp ⏎ (h * spin ε + ∑ j, message (theta j).1 (fun l => x (j, l)) ε)) / 2)", "ComparatorChallenges/DilutedSpin.lean:96-102: local instance instMeasurableSpaceCarrier_challenge (X : TopCat) : MeasurableSpace X := borel X ⏎ local instance instBorelSpaceCarrier_challenge (X : TopCat) : BorelSpace X := ⟨rfl⟩ ⏎ ⏎ /-- Nested probability laws: H_0=R and H_{j+1}=P(H_j), with weak topologies. -/ ⏎ noncomputable def Hierarchy : ℕ → TopCat ⏎ | 0 => TopCat.of ℝ ⏎ | r + 1 => TopCat.of (ProbabilityMeasure (Hierarchy r))", "ComparatorChallenges/DilutedSpin.lean:107-113: noncomputable def logMean {ι : Type} [Fintype ι] : ⏎ (r : ℕ) → ((ι → ℝ) → ℝ) → (Fin r → ℝ) → (ι → Hierarchy r) → ℝ ⏎ | 0, g, _, x => g x ⏎ | r + 1, g, m, eta => ⏎ Real.log (∫ x : ι → Hierarchy r, ⏎ Real.exp (m 0 * logMean r g (fun i => m i.succ) x) ⏎ ∂Measure.pi (fun i => (eta i).toMeasure)) / m 0", "ComparatorChallenges/DilutedSpin.lean:117-124: noncomputable def trialLog {ι : Type} [Fintype ι] (r : ℕ) ⏎ (ζ : Hierarchy (r + 1)) (m : Fin r → ℝ) (g : (ι → ℝ) → ℝ) : ℝ := ⏎ ∫ eta : ι → Hierarchy r, logMean r g m eta ⏎ ∂Measure.pi (fun _ : ι => ζ.toMeasure) ⏎ ⏎ /-- Admissible exponents 0 expectedPressure α β φ (N + 1)) atTop (𝓝 p) ∧" ], "encoding": [ "finite temperature = every β > 0", "iid N(0,1) patterns, ⌊αN⌋ of them; spins uniform on the sphere of radius √N", "pressure = (1/N) log ∫ exp(β Σ_a φ(⟨g_a,x⟩/√N)); convergence in expectation and in probability to a real variational value p" ] }, { "clause": "A spherical extension treats bi-orthogonally invariant disorder with compact limiting singular-value distributions and no outliers.", "status": "unlinked", "coveredBy": [], "manuscript": "The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026", "evidence": [ "CONTENTS.md:5267: We determine the limiting free energy of a spherical perceptron with a bounded continuous activation and a bi-orthogonally invariant disorder matrix.", "lean/docs/222.md:5: - [The free energy of the Ising random perceptron](../../preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026/The-free-energy-of-the-Ising-random-perceptron-September-24-2026.pdf)", "lean/docs/222.md:6: - [The free energy of the spherical random perceptron](../../preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026/The-free-energy-of-the-spherical-random-perceptron-September-24-2026.pdf)" ] }, { "clause": "At margin −1, the quadratic-penalty spherical model has a sharp feasibility threshold and limiting gap and force laws, with system size, zero temperature, and critical density taken in that order.", "status": "unlinked", "coveredBy": [], "manuscript": "Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026", "evidence": [ "CONTENTS.md:5254: We prove a sharp feasibility threshold and limiting gap and force laws for the spherical perceptron with margin −1 and quadratic penalty.", "lean/docs/222.md:6: - [The free energy of the spherical random perceptron](../../preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026/The-free-energy-of-the-spherical-random-perceptron-September-24-2026.pdf)" ] } ], "gap": "Ising: only finiteness of the variational value (no convergence; continuous potentials only) — declared. The bi-orthogonally invariant spherical model and the margin −1 jamming threshold, gap and force laws are not stated, and the scope note is silent on both.", "scopeNoteOnGap": "Ising: lean/docs/222.md:12: It does not assert convergence of the finite-system pressure or the paper's extension to all bounded Borel log-potentials. Bi-orthogonal and jamming clauses: silent (neither paper is linked).", "note": "The spherical free energy is stated in full; the Ising gap is declared by the note; the bi-orthogonal and jamming clauses come only from the two manuscripts the note does not link. So NARROWER — UNLINKED. My clause reading is the first reader's; the word differs only because UNLINKED did not exist for them. SphericalField (a supporting dual formula, not on the README support list) covers no headline clause.", "firstSuspicious": "Pattern seen across this batch: the CONTENTS.md headline was widened to manuscripts the scope note does not link, and the scope note never says those manuscripts are unformalized. Strictly that is silence, hence UNDECLARED; a second reader should weigh whether the scope note's paper list counts as a declaration. Here: the bi-orthogonal and the microscopic-jamming papers.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "IsingFiniteness", "flag": "JUNK?: patternFunctional = Filter.limUnder of a sequence (IsingFiniteness.lean:75); if the sequence did not converge the value is an unspecified choice — this can make the finiteness claim unprovable, not trivially true; supporting result only" }, { "challenge": "SphericalField", "flag": "JUNK?: boundedSphericalFieldValue = limUnder of rounded finite-depth pressures (SphericalField.lean:458); a non-existent limit would give an arbitrary value (unprovable rather than vacuous); supporting result" } ], "flagReadings": null, "readings": [ "first/batch4.json", "second/second4.json" ] }, { "family": "227", "title": "Critical SK autocorrelation processes and dynamics across the temperature transition", "headline": "For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the $`\\log n`$ scale for fixed $`0\\le\\beta\\lt 1`$, mixing time $`n^{2/3+o(1)}`$ at β = 1, and stretched-exponential mixing from a Gibbs-sampled fixed starting configuration for β > 1, in probability over disorder. At criticality, rescaled stationary and quench autocorrelation processes have universal random limits for Gaussian and Rademacher disorder; the quench limit relaxes to the stationary limit.", "challenges": [ "SKHighTemperature", "SKRatio", "CriticalSKMixing", "CriticalSK", "SKBarriers" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the $`\\log n`$ scale for fixed $`0\\le\\beta\\lt 1`$", "status": "declared", "coveredBy": [ "SKRatio" ], "manuscript": "Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026", "evidence": [ "lean/ComparatorChallenges/SKRatio.lean:60: (hβ0 : 0 ≤ β) (hβhalf : β < 1 / 2)", "lean/ComparatorChallenges/SKRatio.lean:65: (mixingTime g ε : ℝ) / (mixingTime g (1 - ε) : ℝ)})", "lean/docs/227.md:14: The formalized supporting result proves ratio cutoff for discrete single-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model for $0\\le\\beta<1/2$.", "lean/docs/227.md:16: The paper's full range $\\beta<1$, explicit cutoff location, and continuous-time conclusion are outside this selected statement." ] }, { "clause": "mixing time $`n^{2/3+o(1)}`$ at β = 1", "part": "lower half: t_mix ≥ n^{2/3−ε} (and n^{5/3−ε} attempts)", "status": "stated", "coveredBy": [ "CriticalSKMixing", "CriticalSK" ], "reading": "critical_mixing_bounds: for every ε > 0, with disorder probability → 1, n^{2/3−ε} ≤ continuous worst-start t_mix(1/4) ≤ e^{εn}, and n^{5/3−ε} ≤ discrete t_mix ≤ e^{εn}.", "evidence": [ "lean/ComparatorChallenges/CriticalSKMixing.lean:69: (n : ℝ) ^ ((2 : ℝ) / 3 - ε) ≤ continuousMixingTime W ∧" ] }, { "clause": "mixing time $`n^{2/3+o(1)}`$ at β = 1", "part": "upper half: t_mix ≤ n^{2/3+ε}", "status": "unlinked", "coveredBy": [], "manuscript": "Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026", "evidence": [ "lean/ComparatorChallenges/CriticalSKMixing.lean:70: continuousMixingTime W ≤ Real.exp (ε * n)}).toReal) atTop (𝓝 1) ∧", "CONTENTS.md:5482: The mixing time is $`n^{2/3+o(1)}`$ when every site has rate one, or $`n^{5/3+o(1)}`$ attempted uniform-site updates, in probability over the disorder.", "lean/docs/227.md:18: the continuous-time mixing time lies between $n^{2/3-\\varepsilon}$ and $e^{\\varepsilon n}$", "lean/docs/227.md:5: - [A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase](../../preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf)", "lean/docs/227.md:6: - [Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase](../../preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf)", "lean/docs/227.md:7: - [Critical slowing down in the Sherrington–Kirkpatrick model](../../preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf)", "lean/docs/227.md:8: - [Stretched-exponential barriers for typical SK initial states](../../preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf)" ] }, { "clause": "and stretched-exponential mixing from a Gibbs-sampled fixed starting configuration for β > 1, in probability over disorder.", "part": "lower half: the barrier at time exp(n^{1/10000})", "status": "stated", "coveredBy": [ "SKBarriers" ], "reading": "SK.main: for every β > 1, the disorder-expected Gibbs mass of starts still at TV > 1/4 at time exp(n^{1/10000}) → 1, continuous and discrete; a [0,1]-valued mass with expectation → 1 tends to 1 in probability.", "evidence": [ "lean/ComparatorChallenges/SKBarriers.lean:63: def timeScale (n : ℕ) : ℝ := Real.exp ((n : ℝ) ^ kappa)", "lean/ComparatorChallenges/SKBarriers.lean:84: theorem main (β : ℝ) (hβ : 1 < β) :", "lean/ComparatorChallenges/SKBarriers.lean:85: Tendsto (continuousBadMass β) atTop (𝓝 1) ∧" ] }, { "clause": "and stretched-exponential mixing from a Gibbs-sampled fixed starting configuration for β > 1, in probability over disorder.", "part": "upper half: mixing by time exp(n^{1−1/40000000}) from a Gibbs-sampled start", "status": "unlinked", "coveredBy": [], "manuscript": "A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026", "evidence": [ "CONTENTS.md:5473: If one Gibbs-sampled configuration is held fixed as the initial state, its total-variation distance from equilibrium is at most 1/4 by time $`\\exp(n^{1-1/40000000})`$", "lean/docs/227.md:22: The formalization proves a stretched-exponential mixing obstruction for the zero-field Sherrington–Kirkpatrick model at every fixed inverse temperature $\\beta>1$.", "lean/docs/227.md:5: - [A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase](../../preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf)", "lean/docs/227.md:6: - [Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase](../../preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf)", "lean/docs/227.md:7: - [Critical slowing down in the Sherrington–Kirkpatrick model](../../preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf)", "lean/docs/227.md:8: - [Stretched-exponential barriers for typical SK initial states](../../preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf)" ] }, { "clause": "At criticality, rescaled stationary and quench autocorrelation processes have universal random limits for Gaussian and Rademacher disorder", "status": "unlinked", "coveredBy": [], "manuscript": "Functional-universality-of-critical-SK-autocorrelations-October-5-2026; Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026", "evidence": [ "CONTENTS.md:5419: We prove joint functional convergence of the stationary and quench autocorrelations of zero-field Sherrington–Kirkpatrick heat-bath dynamics at inverse temperature β = 1, with the same random limit for Gaussian and Rademacher couplings.", "CONTENTS.md:5428: We prove that the stationary spin autocorrelation of the zero-field Sherrington–Kirkpatrick model at inverse temperature β = 1 has a common functional scaling limit for Gaussian and Rademacher couplings.", "lean/docs/227.md:5: - [A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase](../../preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf)", "lean/docs/227.md:6: - [Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase](../../preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf)", "lean/docs/227.md:7: - [Critical slowing down in the Sherrington–Kirkpatrick model](../../preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf)", "lean/docs/227.md:8: - [Stretched-exponential barriers for typical SK initial states](../../preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf)" ] }, { "clause": "the quench limit relaxes to the stationary limit.", "status": "unlinked", "coveredBy": [], "manuscript": "Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026", "evidence": [ "CONTENTS.md:5419: The quench limit is selected by these initial states and relaxes to the stationary limiting autocorrelation as the waiting time tends to infinity.", "lean/docs/227.md:5: - [A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase](../../preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf)", "lean/docs/227.md:6: - [Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase](../../preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf)", "lean/docs/227.md:7: - [Critical slowing down in the Sherrington–Kirkpatrick model](../../preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf)", "lean/docs/227.md:8: - [Stretched-exponential barriers for typical SK initial states](../../preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf)" ] } ], "gap": "Not stated: the critical autocorrelation universality (clause 4, the titular result); the n^{2/3+o(1)} upper bound at β = 1; the stretched-exponential upper bound for β > 1; cutoff for 1/2 ≤ β < 1, its log n location and continuous time (this last part is declared). Stated: the high-temperature spectral gap (supporting), discrete ratio cutoff for β < 1/2, critical lower bounds with an e^{εn} upper bound, the β > 1 barrier, and critical covariance/linear-test scaling.", "scopeNoteOnGap": "Partly declared: lean/docs/227.md:16: \"The paper's full range $\\beta<1$, explicit cutoff location, and continuous-time conclusion are outside this selected statement.\" Silent on the rest. Line 18 states the β = 1 range \"between $n^{2/3-\\varepsilon}$ and $e^{\\varepsilon n}$\" without saying the n^{2/3+o(1)} upper bound is left out; the typical-start upper-bound and autocorrelation manuscripts are not linked (lines 5–8) or mentioned.", "note": "The cutoff clause is covered only for discrete ratio cutoff at β < 1/2, and docs/227.md:16 names the rest (full range, location, continuous time): DECLARED. Every other gap comes from a manuscript the note does not link — the β = 1 upper bound n^{2/3+o(1)} ('Critical mixing', Sept 25; the stated upper bound is only e^{εn}), the β > 1 typical-start upper bound ('A typical-start upper bound'), and both autocorrelation results (the two October-5 papers, which give the family its title): UNLINKED. No gap sits in a linked manuscript without a declaration, so the word is NARROWER — UNLINKED; the first reader's UNDECLARED predates that word. No grep hit for autocorrelation, quench or Rademacher in any of the five challenges.", "firstSuspicious": "The family is titled for the critical autocorrelation processes (October 5 manuscripts), yet no challenge touches them and the scope note never mentions them. The β = 1 upper bound gap is also easy to miss: the note's 'between n^{2/3−ε} and e^{εn}' is accurate but much weaker than n^{2/3+o(1)}.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "SKRatio", "flag": "JUNK? checked, none: the ratio's denominator could be 0 (x/0 = 0), but the second conjunct makes it positive eventually for all disorders." }, { "challenge": "CriticalSKMixing", "flag": "JUNK? checked, none: continuousMixingTime is an sInf of a set bounded below by 0; an empty set would give 0 and falsify the lower bound." }, { "challenge": "CriticalSK", "flag": "JUNK? checked, none: linearRayleigh is a Real.sSup (0 if unbounded) but is bounded below by n^{2/3}/M_n > 0 in the event, so junk cannot satisfy it." }, { "challenge": "SKBarriers", "flag": "JUNK? checked, none: a Bochner integral of a non-integrable function is 0, which would falsify (not trivialize) the limit 1; the integrand is bounded in [0, 1]." } ], "flagReadings": null, "readings": [ "first/batch5.json", "second/second5.json" ] }, { "family": "228", "title": "Continuum phase transitions for radial pair potentials", "headline": "Constructs stable distance-dependent pair interactions for three-dimensional classical particles with a first-order phase transition: the canonical free energy has a derivative jump at one inverse temperature throughout an open density interval. One potential has a divergent repulsive core; another is bounded and continuous with an integrable power-law tail, realizing the type of transition sought in Simon's continuum problem.", "challenges": [ "ContinuumTransition", "RadialTransition", "RadialDensityInterval" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs stable distance-dependent pair interactions for three-dimensional classical particles with a first-order phase transition: the canonical free energy has a derivative jump at one inverse temperature throughout an open density interval.", "coveredBy": [ "ContinuumTransition", "RadialDensityInterval" ], "note": "Both state a common βc with left/right β-derivatives of the free energy satisfying d₊ < d₋ for every density in an open interval." }, { "clause": "One potential has a divergent repulsive core;", "coveredBy": [ "ContinuumTransition" ], "note": "φ → +∞ as r → 0⁺, plus an attractive interval, stability and φ = o(r⁻³)." }, { "clause": "another is bounded and continuous with an integrable power-law tail,", "coveredBy": [ "RadialTransition", "RadialDensityInterval" ], "note": "|φ(r)| ≤ C r^{-3-1/32} for r ≥ 1; RadialTransition at the single density 5p/3, RadialDensityInterval on an interval around it." }, { "clause": "realizing the type of transition sought in Simon's continuum problem.", "coveredBy": [], "note": "Interpretive gloss; nothing to formalize." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "ContinuumTransition", "flag": "JUNK? (not triggered): the partition function is a Bochner integral (0 if not integrable) fed to Real.log (log 0 = 0); Admissible requires Measurable (radialPotential φ) and stability, so the Boltzmann factor is measurable, bounded by e^{βBN} on a bounded cube, integrable and positive" } ], "flagReadings": [ { "challenge": "ContinuumTransition", "flag": "JUNK? (not triggered): the partition function is a Bochner integral (0 if not integrable) fed to Real.log (log 0 = 0); Admissible requires Measurable (radialPotential φ) and stability, so the Boltzmann factor is measurable, bounded by e^{βBN} on a bounded cube, integrable and positive", "decision": "cleared", "why": "φ is existential, so what matters is that every φ satisfying Admissible gets the genuine partition function, and it does. The integral runs over cube(L)^N, a set of finite measure. boltzmann is measurable: radialPotential φ is measurable (Admissible's first conjunct), it is composed with the continuous x ↦ x i − x j, and the terms take values in ℝ ∪ {⊤}, never ⊥. It is ≥ 0, and it is ≤ exp(βBN) because the stability conjunct gives energy ≥ −BN wherever energy is finite (and boltzmann = 0 where energy = ⊤). So it is integrable and the Bochner integral is the true one. For N ≥ 1 and L > 0, configurations with distinct particles have full measure in the box. There the energy is finite and the integrand is > 0, so the partition function is > 0 and Real.log reads the true logarithm. N = 0 gives the defined value 1, and L ≤ 0 lies outside the atTop limit. Lane K: CERTIFIED.", "evidence": [ "lean/ComparatorChallenges/ContinuumTransition.lean:20: Measurable (radialPotential φ) ∧", "lean/ComparatorChallenges/ContinuumTransition.lean:27: (∃ B : ℝ, ∀ (N : ℕ) (x : Fin N → Space),", "lean/ComparatorChallenges/ContinuumTransition.lean:28: ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x)", "lean/ComparatorChallenges/ContinuumTransition.lean:35: if energy φ x = ⊤ then 0 else Real.exp (-β * (energy φ x).toReal)", "lean/ComparatorChallenges/ContinuumTransition.lean:40: ∫ x in {x : Fin N → Space | ∀ i, x i ∈ cube L}, boltzmann φ β x", "lean/ComparatorChallenges/ContinuumTransition.lean:44: Tendsto (fun L : ℝ => L⁻¹ ^ 3 * Real.log (partition φ β L ⌊ρ * L ^ 3⌋₊))" ] } ], "readings": [ "first/batch1.json", "flags/flags1.json" ] }, { "family": "229", "title": "Exact three- and four-state reconstruction thresholds and four-state tree capacity", "headline": "Proves the exact reconstruction threshold $`d\\lambda^2\\gt 1`$, with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees (d ≥ 2) and observed Poisson trees (mean d > 1 and d > 0, respectively), with Poisson advantage averaged without conditioning on survival. The three-state theorem allows both signs of λ and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model.", "challenges": [ "ThreeStateSupercritical", "ThreeStateTreeClauses" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves the exact reconstruction threshold $`d\\lambda^2\\gt 1`$, with nonreconstruction at equality, for three-state symmetric [...] broadcasting on regular trees (d ≥ 2) and observed Poisson trees (mean d > 1 [...]", "status": "declared", "coveredBy": [ "ThreeStateSupercritical", "ThreeStateTreeClauses" ], "manuscript": "The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026", "partial": "the supercritical direction (dλ² > 1 ⇒ advantage → a positive limit) is stated; non-reconstruction at and below the threshold, which makes the threshold exact, is not", "evidence": [ "lean/ComparatorChallenges/ThreeStateSupercritical.lean:98-104: theorem regular_supercritical (b : ℕ) (hb : 2 ≤ b) (lam : ℝ) (h : Admissible lam) ⏎ (hcrit : 1 < (b : ℝ) * lam ^ 2) : Reconstructs (regularAdvantage b lam h) := by ⏎ sorry ⏎ theorem poisson_supercritical (d : ℝ) (hd : 1 < d) (lam : ℝ) (h : Admissible lam) ⏎ (hcrit : 1 < d * lam ^ 2) : Reconstructs (poissonAdvantage d hd lam h) := by ⏎ sorry", "lean/docs/229.md:11: The selected statements cover reconstruction above the Kesten–Stigum threshold. Non-reconstruction at or below the threshold and the stochastic-block-model consequences in the paper are outside them." ] }, { "clause": "[...] and four-state ferromagnetic broadcasting on regular trees (d ≥ 2) and observed Poisson trees ([...] d > 0, respectively)", "status": "unlinked", "coveredBy": [], "manuscript": "The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026", "evidence": [ "preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/build/main.tex:44-48: We establish the exact Kesten--Stigum reconstruction threshold for the ⏎ ferromagnetic four-state Potts broadcast model on every regular $d$-ary ⏎ tree with $d\\ge2$ and every Poisson Galton--Watson tree of mean $d>0$. ⏎ Reconstruction occurs exactly when $d\\lambda^2>1$; we prove ⏎ nonreconstruction at and below the threshold, including equality.", "lean/docs/229.md:5: - [The exact reconstruction threshold for the three-state symmetric channel](../../preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/paper.pdf)", "lean/docs/229.md:9: The formalization proves the supercritical direction of reconstruction for the symmetric three-state broadcast channel, whose parameter satisfies $-1/2\\le\\lambda\\le1$. On a regular $b$-ary tree, reconstruction holds when $b\\lambda^2>1$; on an observed Poisson Galton–Watson tree of mean $d$, it holds when $d\\lambda^2>1$. In each case the root-estimation advantage converges to a positive limit, with the Poisson advantage averaged over trees and spins." ] }, { "clause": "with Poisson advantage averaged without conditioning on survival", "status": "stated", "coveredBy": [ "ThreeStateSupercritical", "ThreeStateTreeClauses" ], "partial": "stated for the three-state channel; for four states it falls with clause 2", "evidence": [ "lean/ComparatorChallenges/ThreeStateSupercritical.lean:68-70: noncomputable def advantage {α : Type*} (law : Spin → PMF α) : ℝ := ⏎ ∑' y, (marginal law y).toReal * ⏎ ((∑ i : Spin, |posterior law y i - 1 / 3|) / 2)", "lean/ComparatorChallenges/ThreeStateSupercritical.lean:83-92: noncomputable def poissonLaw (d : ℝ) (hd : 0 ≤ d) (lam : ℝ) ⏎ (h : Admissible lam) (n : ℕ) := ⏎ observedLaw (poissonOffspring d hd) lam h n ⏎ noncomputable def regularAdvantage (b : ℕ) (lam : ℝ) (h : Admissible lam) (n : ℕ) : ℝ := ⏎ advantage (regularLaw b lam h n) ⏎ noncomputable def poissonAdvantage (d : ℝ) (hd : 1 < d) (lam : ℝ) ⏎ (h : Admissible lam) (n : ℕ) : ℝ := ⏎ advantage (poissonLaw d (le_trans (by norm_num) hd.le) lam h n)" ] }, { "clause": "The three-state theorem allows both signs of λ", "status": "stated", "coveredBy": [ "ThreeStateSupercritical", "ThreeStateTreeClauses" ], "partial": "both signs (−1/2 ≤ λ ≤ 1) in the stated supercritical direction; non-reconstruction for either sign is the declared omission of clause 1", "evidence": [ "lean/ComparatorChallenges/ThreeStateSupercritical.lean:10: def Admissible (lam : ℝ) : Prop := -(1 / 2 : ℝ) ≤ lam ∧ lam ≤ 1" ] }, { "clause": "and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model.", "status": "declared", "coveredBy": [], "manuscript": "The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026", "evidence": [ "lean/docs/229.md:11: The selected statements cover reconstruction above the Kesten–Stigum threshold. Non-reconstruction at or below the threshold and the stochastic-block-model consequences in the paper are outside them." ] } ], "gap": "(a) Non-reconstruction at and below dλ² = 1 (the 'exact' part of the threshold, and the new direction — reconstruction above Kesten–Stigum is classical) is not stated for either model; (b) every four-state ferromagnetic result (threshold on regular and Poisson trees, nonreconstruction at equality, and the L³-capacity criterion) is not stated; (c) the three-community SBM weak-recovery threshold is not stated. (a) for three states and (c) are declared; (a) for four states and (b) are not.", "scopeNoteOnGap": "partly declared: docs/229.md:11 'The selected statements cover reconstruction above the Kesten–Stigum threshold. Non-reconstruction at or below the threshold and the stochastic-block-model consequences in the paper are outside them.' — but 'the paper' is the single three-state manuscript listed at docs/229.md:5; the scope note is silent on the two four-state manuscripts (no mention of four-state, Potts or capacity).", "note": "Only reconstruction above Kesten–Stigum for three states is stated; the note names the non-reconstruction and SBM omissions for the three-state paper it links, and the four-state threshold is the unlinked October-5 paper. Most severe clause is UNLINKED; differs from the first reader only by amendment 3 (their undeclared part was the four-state results).", "firstSuspicious": "Both challenges state only the classical Kesten–Stigum direction (reconstruction when dλ² > 1); the papers' new content — non-reconstruction at and below the threshold — is stated nowhere, and the four-state manuscripts have no challenge and no mention in the scope note.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json", "second/second6.json" ] }, { "family": "230", "title": "Exact Hausdorff gauges for SLE", "headline": "Resolves Schramm’s Hausdorff-measure question for chordal SLEκ, $`0\\lt \\kappa\\lt 8`$. The explicit gauge $`r^d(\\log\\log(1/r))^{(2-d)/2}`$, $`d=1+\\kappa/8`$, gives almost surely positive finite measure to every trace segment $`\\gamma([s,t])`$ with $`0\\lt s\\lt t\\lt \\infty`$, and finite expected measure to the trace in every bounded disk.", "challenges": [ "SLELowerPositivity" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Resolves Schramm’s Hausdorff-measure question for chordal SLEκ, $`0\\lt \\kappa\\lt 8`$.", "status": "undeclared", "coveredBy": [], "manuscript": "An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026", "evidence": [ "CONTENTS.md:5566: This gives an explicit solution to Schramm's Hausdorff-measure problem in this parameter range.", "lean/docs/230.md:6: - [An explicit exact Hausdorff gauge for SLE: A regular formula from quantitative tails and dense visits](../../preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf)", "lean/docs/230.md:10: almost surely assigns positive measure to every nontrivial compact positive-time segment of ordinary chordal $\\mathrm{SLE}_\\kappa$.", "lean/docs/230.md:16: | Positivity of the explicit gauge on every positive-time segment |", "lean/ComparatorChallenges/SLELowerPositivity.lean:76: 0 < hausdorffGauge h (segment (γ ω) s t)" ], "note": "Schramm's question asks for positive AND finite measure; only positivity is stated." }, { "clause": "The explicit gauge $`r^d(\\log\\log(1/r))^{(2-d)/2}`$, $`d=1+\\kappa/8`$, gives almost surely positive", "status": "stated", "coveredBy": [ "SLELowerPositivity" ], "evidence": [ "lean/ComparatorChallenges/SLELowerPositivity.lean:57: r ^ dimension κ * (Real.log (Real.log (1 / r))) ^ (codimension κ / ((2 : ℕ) : ℝ))", "lean/ComparatorChallenges/SLELowerPositivity.lean:74: ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →", "lean/ComparatorChallenges/SLELowerPositivity.lean:75: ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →", "lean/ComparatorChallenges/SLELowerPositivity.lean:76: 0 < hausdorffGauge h (segment (γ ω) s t)" ], "note": "POSITIVE half only, a.s. simultaneously for all 0 < s < t, for every continuous nondecreasing gauge agreeing with the formula near 0+. SLE is encoded as a Loewner trace driven by √κ·B (capacity-two witness); the gauge measure is Measure.mkMetric, whose value at infinite diameter cannot raise a lower bound." }, { "clause": "finite measure to every trace segment $`\\gamma([s,t])`$", "status": "undeclared", "coveredBy": [], "manuscript": "An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026", "evidence": [ "CONTENTS.md:5566: almost surely gives positive finite Hausdorff measure to every nontrivial positive-time compact segment of chordal SLEκ.", "lean/docs/230.md:6: - [An explicit exact Hausdorff gauge for SLE: A regular formula from quantitative tails and dense visits](../../preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf)", "lean/docs/230.md:10: almost surely assigns positive measure to every nontrivial compact positive-time segment of ordinary chordal $\\mathrm{SLE}_\\kappa$.", "lean/docs/230.md:16: | Positivity of the explicit gauge on every positive-time segment |" ], "note": "No upper bound on hausdorffGauge appears in the challenge." }, { "clause": "and finite expected measure to the trace in every bounded disk.", "status": "undeclared", "coveredBy": [], "manuscript": "An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026", "evidence": [ "CONTENTS.md:5566: The entire trace has finite expected measure in every bounded disk.", "CONTENTS.md:5557: The entire trace has finite expected gauge measure in each bounded box.", "lean/docs/230.md:6: - [An explicit exact Hausdorff gauge for SLE: A regular formula from quantitative tails and dense visits](../../preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf)", "lean/docs/230.md:10: almost surely assigns positive measure to every nontrivial compact positive-time segment of ordinary chordal $\\mathrm{SLE}_\\kappa$.", "lean/docs/230.md:16: | Positivity of the explicit gauge on every positive-time segment |" ] } ], "gap": "finiteness of the gauge measure on segments and finite expected measure in bounded disks (the upper half of the exact-gauge claim)", "scopeNoteOnGap": "silent — docs/230.md:10 states only positivity (\"... almost surely assigns positive measure to every nontrivial compact positive-time segment of ordinary chordal $\\mathrm{SLE}_\\kappa$.\") and never says finiteness is not formalized", "note": "Only the lower (positivity) half of the exact-gauge theorem is stated; finiteness on segments, finite expected measure in disks, and hence the resolution of Schramm's question are not, although both manuscripts claiming them are linked and the note only describes the positive half (its table title says \"Positivity\", which describes the selection without naming the omission): NARROWER — UNDECLARED. Agree with the first reader.", "firstSuspicious": "The challenge and scope-note titles say 'Positivity' / 'LowerPositivity', which signals the selection but does not say finiteness is left out; under the strict rule this is UNDECLARED. Positivity alone does not resolve Schramm's question.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "SLELowerPositivity", "flag": "VACUITY? cleared: SLE exists for 0<κ<8 (Rohde–Schramm), so the hypothesis IsOrdinaryChordalSLE is satisfiable" } ], "flagReadings": null, "readings": [ "first/batch8.json", "second/second7.json" ] }, { "family": "231", "title": "The free uniform spanning forest is a factor of IID", "headline": "On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root. Translation-invariant strongly Rayleigh binary processes on every countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, are also factors of IID.", "challenges": [ "StronglyRayleighDPP", "FreeUniformSpanningForest" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root.", "coveredBy": [ "FreeUniformSpanningForest" ], "note": "∃ one Borel, relabeling-equivariant rule working for every connected locally finite graph on ℕ." }, { "clause": "Translation-invariant strongly Rayleigh binary processes on every countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, are also factors of IID.", "coveredBy": [ "StronglyRayleighDPP" ], "note": "strongly_rayleigh_group_factor, invariant_positive_contraction_dpp_factor, translation_invariant_kernel_dpp_factor." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch1.json" ] }, { "family": "234", "title": "All-temperature pressure of orthogonally invariant Ising spin glasses", "headline": "Gives an exact variational formula for the limiting pressure of orthogonally invariant Ising spin glasses at every fixed temperature, both almost surely and in expectation. The coupling matrix is a Haar-random rotation of a deterministic spectrum converging to a compactly supported law, with extreme eigenvalues converging to its support edges. The zero-field ground-state energy follows as temperature tends to zero.", "challenges": [ "InvariantIsing" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Gives an exact variational formula for the limiting pressure of orthogonally invariant Ising spin glasses at every fixed temperature, both almost surely and in expectation.", "coveredBy": [ "InvariantIsing" ], "note": "positive_temperature_pressure_unconditional (every β>0) and limiting_pressure_of_extreme_limits_unconditional, each asserting convergence in expectation and almost surely to (variationalFunctional …).toReal." }, { "clause": "The coupling matrix is a Haar-random rotation of a deterministic spectrum converging to a compactly supported law, with extreme eigenvalues converging to its support edges.", "coveredBy": [ "InvariantIsing" ], "note": "Hypotheses: U_N Haar (right-invariant law on O(N)), empirical spectral law → ν weakly, supp ν compact ⊆ [a,b] with a,b ∈ supp ν, min/max eigenvalues → a,b (or no outliers)." }, { "clause": "The zero-field ground-state energy follows as temperature tends to zero.", "coveredBy": [ "InvariantIsing" ], "note": "ground_state_limit_unconditional: the ground-state energy converges (mean and a.s.) to M and thermalVariationalValue ν b β → M as β → ∞." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": "496-line statement whose conclusion is a bespoke variational functional; the reading checked structure, hypotheses and junk paths, not term-by-term equality with the manuscript's formula. Also: run_cmd environment edits in the statement file.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [ { "challenge": "InvariantIsing", "flag": "STATEMENT-FILE METAPROGRAM: eight `run_cmd Lean.modifyEnv … auxLemmasExt.setState env {}` lines reset Lean's auxiliary-lemma cache inside the statement; they add no declarations, but environment-modifying code in a statement file deserves lane K's attention" }, { "challenge": "InvariantIsing", "flag": "JUNK? (not triggered): limits are (variationalFunctional …).toReal and EReal.toReal sends ±∞ to 0; since every theorem asserts convergence of genuine bounded pressures (or of the genuine ground-state energy, which pins M) to that value, a junk value would make the statement false, not weaker" }, { "challenge": "InvariantIsing", "flag": "DEFINITION carries the formula: measureR (R-transform) uses h.choose with fallback e (so R(x) = e - 1/x beyond the edge) and R(x) := mean for x ≤ 0; entropyFunctional is a ⨆ over finite FieldSteps. Reading confirms the Parisi/R-transform structure but cannot confirm these equal the manuscript's formula; a mismatch would make the theorems false rather than vacuous" } ], "flagReadings": [ { "challenge": "InvariantIsing", "flag": "JUNK? (not triggered): limits are (variationalFunctional …).toReal and EReal.toReal sends ±∞ to 0; since every theorem asserts convergence of genuine bounded pressures (or of the genuine ground-state energy, which pins M) to that value, a junk value would make the statement false, not weaker", "decision": "cleared", "why": "The flagged mechanism is EReal.toReal, which sends ⊤ and ⊥ to 0. Under the theorems' hypotheses the EReal is finite, so toReal is faithful. (i) It is > ⊥. entropyFunctional p ≥ 0 for every p, because the depth-0 FieldStep with cut 0 = 0, cut 1 = 1 and height 0 gives fieldValue = log cosh 0 − 0 = 0 and fieldPairing = 0. spectralFunctional is a real number. For p valued in [0,1], deficit p r ∈ [0, 1−r], and on that range measureR of a compactly supported law with upper edge e is bounded: it is G⁻¹(x) − 1/x below G(e+), which tends to the mean as x → 0+, and e − 1/x above. So the spectral term is uniformly bounded below. (ii) It is < ⊤. At p ≡ 0 the Parisi-type recursion has every a = cut ∈ [0,1], so it is dominated by the a = 1 recursion (Jensen at a = 0, the power-mean inequality for 0 < a ≤ 1). The a = 1 recursion telescopes to log E cosh(√V g) = V/2, which the −height_last/2 = −V/2 term cancels. So entropyFunctional 0 ≤ 0, and variationalFunctional ≤ spectralFunctional R 0. Independently, every conclusion equates a model-determined quantity to the value: pressure integrals, a.e. pressures, Y tied to the physical pressure by ConditionalFieldOrbitLaw, and the ground-state energies or norms that pin the existential M, e, eplus and eminus. A junk 0 would therefore make a statement false, not weaker. Aside (not this flag): magneticFieldFunctional is a limUnder (lines 193-195), which is junk if the approximating sequence diverges. It also appears only as the limit of a genuine sequence, so it can only falsify.", "evidence": [ "lean/ComparatorChallenges/InvariantIsing.lean:72: 1 - r - ∫ s, max (p s - r) 0 ∂pathMeasure", "lean/ComparatorChallenges/InvariantIsing.lean:107: ⨆ h : FieldStep, ((fieldValue h 0 + fieldPairing p h / 2 : ℝ) : EReal)", "lean/ComparatorChallenges/InvariantIsing.lean:110: (1 / 2 : ℝ) * ∫ r, R (deficit p r) ∂pathMeasure", "lean/ComparatorChallenges/InvariantIsing.lean:113: ⨅ p : OverlapPath, entropyFunctional p + (spectralFunctional R p : EReal)", "lean/ComparatorChallenges/InvariantIsing.lean:121: if 0 < x then measureInverse μ e x - 1 / x else ∫ y, y ∂μ", "lean/ComparatorChallenges/InvariantIsing.lean:194: limUnder atTop (fun n => simpleFieldValue (field : Measure ℝ) (fieldSimpleApproximation n)", "lean/ComparatorChallenges/InvariantIsing.lean:295: atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) ∧", "lean/ComparatorChallenges/InvariantIsing.lean:296: ∀ᵐ ω ∂P, Tendsto (fun N => rotatedPressure (eig N) (matrixRotation (U N ω)⁻¹) (fun _ => 0))", "lean/ComparatorChallenges/InvariantIsing.lean:297: atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) := by", "lean/ComparatorChallenges/InvariantIsing.lean:354: Tendsto (thermalVariationalValue ν b) atTop (𝓝 M) := by" ] } ], "readings": [ "first/batch1.json", "flags/flags2.json" ] }, { "family": "235", "title": "Limiting random SAT thresholds, sharp variance and computability", "headline": "For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance $`\\Theta_k(n)`$ for every fixed k ≥ 3, and computability of the 3-SAT threshold. We credit Gaia Carenini with priority for resolving the threshold-existence conjecture in her concurrent [ECCC TR26-229](https://eccc.weizmann.ac.il/report/2026/229/), made public October 5, 2026; this family supplies another proof and the sharper variance and computability results.", "challenges": [ "FixedClauseThreshold", "SATSharpness", "SATVariance", "SATComputability" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds", "coveredBy": [ "FixedClauseThreshold" ], "note": "every k>=3: ∃ α>0, P(SAT) → 1 for densities c<α and → 0 for c>α" }, { "clause": "and hitting-time variance $`\\Theta_k(n)`$ for every fixed k ≥ 3", "coveredBy": [ "SATVariance" ], "note": "Θ_k(n) only for k>=4; for k=3 the Lean upper bound is C·n·log(e·n), i.e. O(n log n), not O(n) — the October-5 linear-variance result is not stated" }, { "clause": "and computability of the 3-SAT threshold.", "coveredBy": [ "SATComputability" ], "note": "one Nat.Partrec.Code computes rationals q_r with |q_r - α| <= 2^-r, α the 3-SAT threshold" } ], "gap": "the k = 3 linear upper bound Var(H_n) = O(n) (Lean has O(n log n))", "scopeNoteOnGap": "The formalization proves $\\mathrm{Var}(H_n)=\\Theta_k(n)$ for every fixed $k\\ge4$. For $k=3$, it proves a positive linear lower bound and an $O(n\\log n)$ upper bound. (docs/235.md:13)", "note": null, "firstSuspicious": "The scope note links only three of the four manuscripts; the October-5 'Linear-Variance-of-the-Random-3-SAT-Hitting-Time' result is exactly the missing k=3 clause, and the scope note does state the O(n log n) bound for k=3.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "SATVariance", "flag": "JUNK? (guarded): Mathlib's variance is 0 when evariance = ∞, which would make an upper bound free; the conjunct c·n <= Var(H_n) for all large n rules this out for large n, and H has exponential tails, so the upper bounds are genuine. Also H uses (firstFailure).toNat, which is 0 on the null event of never failing" }, { "challenge": "SATSharpness", "flag": "JUNK? cleared: lifetime is a tsum (0 if not summable), but it appears only on the larger side of a strict inequality, so junk could not make the claim easier" } ], "flagReadings": [ { "challenge": "SATVariance", "flag": "JUNK? (guarded): Mathlib's variance is 0 when evariance = ∞, which would make an upper bound free; the conjunct c·n <= Var(H_n) for all large n rules this out for large n, and H has exponential tails, so the upper bounds are genuine. Also H uses (firstFailure).toNat, which is 0 on the null event of never failing", "decision": "cleared", "why": "Mathlib's variance is (evariance).toReal, which is 0 when evariance = ∞, and that WOULD make the upper bounds free. The first reader's guard (the lower bound c·n ≤ Var for large n) covers only large n. The decisive reason is different: Var(H_n) is finite for EVERY n ≥ k. A uniformly random proper k-clause kills a fixed assignment with probability exactly 2^(−k), because its signs are uniform. So P(H_n > m) ≤ P(prefix of m clauses satisfiable) ≤ 2^n·(1−2^(−k))^m: H_n has an exponential tail, so it is integrable (the Bochner mean is genuine) and evariance < ∞. T B is bounded by cap n B, so its variance is finite trivially. The toNat-of-⊤ = 0 case is the event that every prefix stays satisfiable, which has probability 0 by the same bound, so it does not affect the variance. No junk is reached. Lane K: CERTIFIED.", "evidence": [ "ComparatorChallenges/SATVariance.lean:26: def Satisfies {n k : ℕ} (c : Clause n k) (a : Assignment n) : Prop :=", "ComparatorChallenges/SATVariance.lean:27: ∃ v : c.1.1, a v = c.2 v", "ComparatorChallenges/SATVariance.lean:39: def firstFailure {n k : ℕ} (ω : Stream n k) : ℕ∞ :=", "ComparatorChallenges/SATVariance.lean:40: sInf ((fun m : ℕ => (m : ℕ∞)) '' {m | ¬ PrefixSAT ω m})", "ComparatorChallenges/SATVariance.lean:41: ", "ComparatorChallenges/SATVariance.lean:42: def H {n k : ℕ} (ω : Stream n k) : ℝ := (firstFailure ω).toNat", "ComparatorChallenges/SATVariance.lean:46: def T {n k : ℕ} (B : ℝ) (ω : Stream n k) : ℝ :=", "ComparatorChallenges/SATVariance.lean:47: (min (firstFailure ω) (cap n B : ℕ∞)).toNat", "ComparatorChallenges/SATVariance.lean:54: theorem variance_main (k : ℕ) (hk : 3 ≤ k) :", "ComparatorChallenges/SATVariance.lean:55: (∃ C : ℝ, 0 ≤ C ∧ ∀ n : ℕ, k ≤ n →", "ComparatorChallenges/SATVariance.lean:56: variance (H : Stream n k → ℝ) (streamLaw n k) ≤ C * n * ell k n) ∧", "ComparatorChallenges/SATVariance.lean:57: (∀ B : ℝ, 0 < B → ∃ C : ℝ, 0 ≤ C ∧ ∀ n : ℕ, k ≤ n →", "ComparatorChallenges/SATVariance.lean:58: variance (T B : Stream n k → ℝ) (streamLaw n k) ≤ C * n * ell k n) ∧", "ComparatorChallenges/SATVariance.lean:59: (∃ c : ℝ, 0 < c ∧", "ComparatorChallenges/SATVariance.lean:60: (∃ N : ℕ, ∀ n : ℕ, N ≤ n → k ≤ n →", "ComparatorChallenges/SATVariance.lean:61: c * n ≤ variance (H : Stream n k → ℝ) (streamLaw n k)) ∧", "ComparatorChallenges/SATVariance.lean:62: (∀ B : ℝ, U k < B → ∃ N : ℕ, ∀ n : ℕ, N ≤ n → k ≤ n →", "ComparatorChallenges/SATVariance.lean:63: c * n ≤ variance (T B : Stream n k → ℝ) (streamLaw n k))) := by" ] }, { "challenge": "SATSharpness", "flag": "JUNK? cleared: lifetime is a tsum (0 if not summable), but it appears only on the larger side of a strict inequality, so junk could not make the claim easier", "decision": "cleared", "why": "The only junk route is the real tsum `lifetime` returning 0. It is the strict right-hand side of D·blockKill^(−γ) < lifetime, whose left side is positive (D > 0, and blockKill ≥ A·u^(−(k−1)) > 0 makes the rpow positive), so a junk 0 could only falsify. It is not reached anyway: survival u k S m ≤ card S·(1−2^(−k))^m (each assignment dies with probability 2^(−k) per clause), so the series converges. Lane K: not run yet.", "evidence": [ "ComparatorChallenges/SATSharpness.lean:53: def survival (u s : ℕ) (S : Finset (Assignment u)) (m : ℕ) : ℝ :=", "ComparatorChallenges/SATSharpness.lean:54: (clauseStep u s)^[m] (alive u) S", "ComparatorChallenges/SATSharpness.lean:55: ", "ComparatorChallenges/SATSharpness.lean:56: def lifetime (u s : ℕ) (S : Finset (Assignment u)) : ℝ := ∑' m, survival u s S m", "ComparatorChallenges/SATSharpness.lean:59: (∀ γ D A : ℝ, γ < (k : ℝ) / ((k-1 : ℕ) : ℝ) → 0 < D → 0 < A →", "ComparatorChallenges/SATSharpness.lean:60: ∀ N : ℕ, ∃ u : ℕ, N ≤ u ∧ k ≤ u ∧ ∃ S : Finset (Assignment u),", "ComparatorChallenges/SATSharpness.lean:61: S.Nonempty ∧ A * (u : ℝ) ^ (-((k-1 : ℕ) : ℝ)) ≤ blockKill u (k-1) S 1 ∧", "ComparatorChallenges/SATSharpness.lean:62: D * (blockKill u (k-1) S 1) ^ (-γ) < lifetime u k S) ∧" ] } ], "readings": [ "first/batch8.json", "flags/flags3.json" ] }, { "family": "236", "title": "The exact factor-of-IID threshold for free Ising spins on trees", "headline": "Determines when the free zero-field ferromagnetic Ising state on the infinite d-regular tree is a factor of independent vertex labels: exactly when $`\\tanh\\beta\\le(d-1)^{-1/2}`$, including equality, for d ≥ 3 and β ≥ 0. The construction uses no root and is almost surely equivariant for each fixed tree automorphism, resolving the ferromagnetic case of Lyons's question.", "challenges": [ "FreeIsing" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "is a factor of independent vertex labels: exactly when $`\\tanh\\beta\\le(d-1)^{-1/2}`$, including equality, for d ≥ 3 and β ≥ 0", "coveredBy": [ "FreeIsing" ], "note": "Iff in both directions for every d ≥ 3, β ≥ 0." }, { "clause": "The construction uses no root and is almost surely equivariant for each fixed tree automorphism", "coveredBy": [ "FreeIsing" ], "note": "∀ automorphism g, a.s. Φ(g·U) = g·Φ(U)." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "237", "title": "The three-quarter exponent for honeycomb self-avoiding walk", "headline": "Proves the diameter form of Nienhuis's predicted three-quarter exponent: a uniformly chosen n-step self-avoiding walk on the honeycomb lattice has diameter $`n^{3/4+o(1)}`$. Its local mass and covering numbers have exponent 4/3. These estimates hold at every sufficiently large fixed length, simultaneously across scales, with arbitrarily high polynomial probability.", "challenges": [ "HoneycombBridgeFiniteness", "HoneycombFreeEnergy", "CriticalStripMass", "HoneycombBridgeMassSupport" ], "word": "DIFFERENT", "firstWord": "DIFFERENT", "secondWord": "DIFFERENT", "disagreement": null, "clauses": [ { "clause": "Proves the diameter form of Nienhuis's predicted three-quarter exponent: a uniformly chosen n-step self-avoiding walk on the honeycomb lattice has diameter $`n^{3/4+o(1)}`$.", "status": "declared", "coveredBy": [], "manuscript": "Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026", "evidence": [ "lean/docs/237.md:18: It does not state the paper's small-force exponent, near-critical correlation scale, or the $3/4$ spatial and moment laws.", "corpus/openai-math/release.json:10314: \"dir\": \"Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026\"", "corpus/openai-math/release.json:10316: For uniform self-avoiding walks of every sufficiently large integer length on the honeycomb lattice, we prove diameter exponent 3/4", "lean/docs/237.md:5: - [Polynomial vacuum representations and bridge mass for honeycomb walks](../../preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf)", "lean/docs/237.md:6: - [Renewal and changes of law for critical honeycomb walks](../../preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf)", "lean/docs/237.md:7: - [Critical strip-crossing mass on the honeycomb lattice](../../preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf)", "lean/docs/237.md:8: - [Critical honeycomb chords with prescribed boundary endpoints](../../preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026/main.pdf)" ] }, { "clause": "Its local mass and covering numbers have exponent 4/3.", "status": "unlinked", "coveredBy": [], "manuscript": "Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026", "evidence": [ "corpus/openai-math/release.json:10316: simultaneous local-mass and covering exponent 4/3", "lean/docs/237.md:14: The paper's $h^{4/3+o(1)}$ mean-length law and its central-visit and hexagonal-chord exponents are outside them.", "lean/docs/237.md:5: - [Polynomial vacuum representations and bridge mass for honeycomb walks](../../preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf)", "lean/docs/237.md:6: - [Renewal and changes of law for critical honeycomb walks](../../preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf)", "lean/docs/237.md:7: - [Critical strip-crossing mass on the honeycomb lattice](../../preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf)", "lean/docs/237.md:8: - [Critical honeycomb chords with prescribed boundary endpoints](../../preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026/main.pdf)" ] }, { "clause": "These estimates hold at every sufficiently large fixed length, simultaneously across scales, with arbitrarily high polynomial probability.", "status": "unlinked", "coveredBy": [], "manuscript": "Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026", "evidence": [ "corpus/openai-math/release.json:10316: with arbitrary positive exponent slack and arbitrary polynomial failure probability.", "lean/docs/237.md:5: - [Polynomial vacuum representations and bridge mass for honeycomb walks](../../preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf)", "lean/docs/237.md:6: - [Renewal and changes of law for critical honeycomb walks](../../preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf)", "lean/docs/237.md:7: - [Critical strip-crossing mass on the honeycomb lattice](../../preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf)", "lean/docs/237.md:8: - [Critical honeycomb chords with prescribed boundary endpoints](../../preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026/main.pdf)" ] } ], "gap": "The headline (diameter n^{3/4+o(1)} and mass/covering exponent 4/3 for uniform fixed-length honeycomb SAW, the content of 'Mass-and-covering-exponents-for-fixed-length-honeycomb-walks') is stated by no challenge. The four challenges state ingredients from other manuscripts of the family: finiteness of critical bridge sums, existence of the forced free energy, the strip-crossing mass ≍ N^{-1/4} with displacement moment ≍ N^{3/4}, and strict-bridge mass ≍ (1+h)^{-1/4}. None implies the headline and the headline implies none of them.", "scopeNoteOnGap": "partly declared: docs/237.md:18 'This selected result supplies the limiting free energy. It does not state the paper's small-force exponent, near-critical correlation scale, or the $3/4$ spatial and moment laws.' and docs/237.md:14 'These are supporting summability statements. The paper's $h^{4/3+o(1)}$ mean-length law and its central-visit and hexagonal-chord exponents are outside them.' — but the scope note links only four of the thirteen manuscripts (lines 5–8), not the fixed-length mass-and-covering paper that carries the headline, and never says the uniform-SAW diameter / mass / covering exponents are unformalized.", "note": "The whole headline is the Mass-and-covering manuscript, which the note does not link; the four challenges state ingredients (bridge-sum finiteness, free energy, strip mass ~ N^{-1/4} with displacement moment ~ N^{3/4}, strict-bridge mass ~ (1+h)^{-1/4}) that neither imply nor follow from it, so DIFFERENT. Clause 1 is recorded declared because 237.md:18 names \"the 3/4 spatial ... laws\" as not stated (phrased for the linked Renewal paper, whose 3/4 laws are weaker than the headline's every-length form). Only HoneycombBridgeMassSupport is on the README list, so not SUPPORT-ONLY. Agree with the first reader. Lane-K note: HoneycombBridgeMassSupport.lean:71 docstring reads \"Trusted reference hole only; missing necessary main obligation.\"", "firstSuspicious": "Thirteen manuscripts, four challenges, none touching the headline. DIFFERENT is the logical word (ingredients neither imply nor are implied by the headline); because the scope note openly calls two of them supporting and says the 3/4 laws are not stated, a page could also read this as NARROWER — DECLARED. HoneycombBridgeMassSupport's docstring reads 'Trusted reference hole only; missing necessary main obligation.' (line 71).", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json", "second/second1.json" ] }, { "family": "238", "title": "Optimal logarithmic mixing of the Thorp shuffle", "headline": "Proves that the Thorp shuffle randomizes $`N=2^d`$ labeled cards in $`\\Theta(\\log N)`$ physical shuffles, settling its optimal mixing order for power-of-two deck sizes. Convergence is in total variation from the worst initial ordering and concerns the entire permutation, not just individual card positions.", "challenges": [ "ThorpFirstReciprocal", "ThorpRemaining", "PartialPermutation", "ThorpRouting", "OccupiedOverlap", "WeightedSweepMoments", "CoordinateTrace", "ThorpWeightedCompatibility", "SpinAngle", "SignedSweepMoment", "BinarySweep", "CoordinateSweeps", "FourRowPermanent" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that the Thorp shuffle randomizes $`N=2^d`$ labeled cards in $`\\Theta(\\log N)`$ physical shuffles, settling its optimal mixing order for power-of-two deck sizes.", "coveredBy": [ "ThorpRemaining" ], "note": "OptimalOrderMain: IsTheta atTop (mixingTime d) (log 2^d), mixingTime d = sInf{t | TV(law after t Thorp steps, uniform) ≤ 1/4}. Upper-bound pieces also in ThorpRemaining (FrameMain 32800d, InformationMain 2048d), CoordinateTrace and ThorpRouting (Casimir block)." }, { "clause": "Convergence is in total variation from the worst initial ordering and concerns the entire permutation, not just individual card positions.", "coveredBy": [ "ThorpRemaining" ], "note": "The law is on the full permutation group Perm({0,1}^d). FrameMain quantifies ∀ start; mixingTime is defined from the identity start, which equals the worst start because TV to uniform is invariant under right translation for a group random walk (stated for t = 2048d in InformationMain's last conjunct)." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": "The family word rests on ThorpRemaining.OptimalOrderMain alone (no definition holes there). OccupiedOverlap's definition hole covers its entire statement — Comparator by itself certifies nothing about that challenge; lane K must check the bodies before it counts.", "confidence": "high", "conditionalOnLaneK": [ { "challenge": "OccupiedOverlap", "strict": "BODIES DIFFER" }, { "challenge": "SpinAngle", "strict": "BODIES DIFFER" } ], "flags": [ { "challenge": "ThorpRemaining", "flag": "META: run_cmd blocks (lines 112–117, 410, 440, 450, 467, 479) reset Lean.Meta.auxLemmasExt and the matcher cache inside the statement file — bookkeeping from concatenated sources, no semantic effect found by reading, but it is code run while elaborating the trusted statement (lane K)" }, { "challenge": "ThorpRemaining", "flag": "JUNK (harmless direction): mixingTime is sInf of a set of ℕ, 0 if empty; that would make IsTheta false, not trivially true" }, { "challenge": "PartialPermutation", "flag": "JUNK (harmless direction): if the sup were unbounded, Real.sSup would be 0 and the bound would read ≤ 0 — false, not trivial" }, { "challenge": "ThorpRouting", "flag": "META: eight run_cmd blocks resetting auxLemmasExt and the matcher cache (lines 292, 629, 898, 1167, 1533, 1656, 1994, 2294)" }, { "challenge": "ThorpRouting", "flag": "JUNK (low): levelScale n 0 = 0·(1 + log(n/0)) = 0 via x/0 = 0 and log 0 = 0; only reached in the Adaptive block for the trivial representation, where the bounds read ≤ 1 and hold — no trivialization" }, { "challenge": "OccupiedOverlap", "flag": "DEFINITION HOLE: definition_names = IsUnitary, Specht.hilbertSpace, Specht.unitaryRepresentation, Specht.relabelledUnitary and RowColumn.OccupiedOverlapEndpoint. The last is the ENTIRE Prop both theorems prove; Comparator checks only that a hole has the same type (Prop), so a solution with OccupiedOverlapEndpoint := True would pass Comparator alone. hilbertSpace := ⊥ or IsUnitary := False would also empty the statement. Lane K must compare the bodies." }, { "challenge": "WeightedSweepMoments", "flag": "META: run_cmd auxLemmasExt reset (line 246)" }, { "challenge": "CoordinateTrace", "flag": "META: run_cmd auxLemmasExt reset (line 333)" }, { "challenge": "SpinAngle", "flag": "DEFINITION HOLE: definition_names = TypeAngle.row, TypeAngle.column, TypeAngle.phi — the fibre maps used in the hypotheses hH, hK and in the block projections, and the missing-cell factor in conjuncts 1–2; Comparator checks only their types (W → R, W → C, Finset → ℝ). Lane K must compare the bodies." }, { "challenge": "CoordinateSweeps", "flag": "VACUITY? (low): all_grid_moment is conditional on ScaleLarge, SparseScalar.ScaleReady, MainSizeReady for every allowed grid and SparsePerturb; read as jointly satisfiable for large r (finitely many grids with b ≤ 12800 for SparsePerturb; size conditions hold for large sizes); conditional_main is unconditional" } ], "flagReadings": [ { "challenge": "CoordinateSweeps", "flag": "VACUITY? (low): all_grid_moment is conditional on ScaleLarge, SparseScalar.ScaleReady, MainSizeReady for every allowed grid and SparsePerturb; read as jointly satisfiable for large r (finitely many grids with b ≤ 12800 for SparsePerturb; size conditions hold for large sizes); conditional_main is unconditional", "decision": "cleared", "why": "The hypotheses of all_grid_moment are jointly satisfiable, so it is not vacuous. ScaleLarge r needs r·log 2 ≥ (2/e0)² = 4·10⁸, which holds for large r. ScaleReady r needs r·log 2 ≥ 8(3 log 2 + 2(K0+1+log 4))/θ ≈ 2.6·10⁹, plus SizeReady s for every s ≥ 2^r. Each SizeReady clause (log s ≤ s^{θ/4}, 2 ≤ s^{θ/4}, B0+1 ≤ s^{θ/2}, the count, absorb, η, prefactor (needs log s ≳ 4·10¹⁹) and logtwo clauses) holds for all s past a finite threshold and stays true, so r ≳ 10²⁰ suffices. MainSizeReady q (G.size) holds for every allowed G because G.size ≥ 2^r (b ≥ 1, bits ≥ r) and each of its four clauses holds for all large s. SparsePerturb r zStar: slack(G.size) > 0 since G.size > 1; at z = 0 every eventDistance is 0, since choiceWeight 0 > 0 makes each denominator positive; eventDistance is continuous in z near 0. There are finitely many grids (b ≤ 12800, r ≤ bits ≤ 2r) and finitely many nonempty E ⊆ G.Choices, so some δ > 0 works uniformly, and zStar = min(δ, baseZeta r) > 0 qualifies. conditional_main has no hypotheses, so it is unaffected.", "evidence": [ "ComparatorChallenges/CoordinateSweeps.lean:242-243: def ScaleLarge (r : ℕ) : Prop := ⏎ 1 ≤ r ∧ (2/e0)^2 ≤ (r : ℝ)*Real.log 2 ∧ 4/e0 ≤ (r : ℝ)*Real.log 2", "ComparatorChallenges/CoordinateSweeps.lean:245-246: def baseWidth (r : ℕ) : ℝ := ((2^(2*r)).factorial : ℕ) ⏎ def baseZeta (r : ℕ) : ℝ := 1/(4*(baseWidth r)^2)", "ComparatorChallenges/CoordinateSweeps.lean:250-254: structure MainSizeReady (q : ℕ) (s : ℝ) : Prop where ⏎ one : 1 ≤ Real.log s ⏎ sparse : (100004:ℝ)*s^(1-a/4) ≤ s^(1-(1:ℝ)/1600) ⏎ hook : (((2*q+1:ℕ):ℝ)*4+82*c0)*2*s^((4:ℝ)/5+2*a)*Real.log (s+1) ≤ s^((9:ℝ)/10) ⏎ removed : s^((9:ℝ)/10)+s^(1-a/2)*Real.log s ≤ s^(1-a/4)/(20*Real.sqrt (Real.log s))", "ComparatorChallenges/CoordinateSweeps.lean:272-287: structure SizeReady (s : ℝ) : Prop where ⏎ two : 2 ≤ s ⏎ logpow : Real.log s ≤ s^(theta/4) ⏎ powtwo : 2 ≤ s^(theta/4) ⏎ powB : (B0+1:ℝ) ≤ s^(theta/2) ⏎ count : 4/s^(1/(4*(B0:ℝ))) ≤ theta/(64*B0) ⏎ absorb : Real.log (1+2*(B0:ℝ))+Real.log 2 ≤ xi*Real.log s ⏎ eta : s^(-xi) ≤ 1/8 ⏎ prefactor : uniformRate ≤ xi*Real.log s/16 ⏎ logtwo : Real.log 2 ≤ xi*Real.log s/8 ⏎ ⏎ structure ScaleReady (r : ℕ) : Prop where ⏎ two : 2 ≤ r ⏎ logone : 1 ≤ (r:ℝ)*Real.log 2 ⏎ absorb : 3*Real.log 2+2*(K0+1+Real.log 4) ≤ theta*((r:ℝ)*Real.log 2)/8 ⏎ size : ∀ s : ℝ, (2:ℝ)^r ≤ s → SizeReady s", "ComparatorChallenges/CoordinateSweeps.lean:289-290: def slack (s : ℝ) : ℝ := ⏎ if 1 < s then s ^ (-((2 : ℕ) : ℝ) * rho * s) * (s ^ rho - 1) else 1", "ComparatorChallenges/CoordinateSweeps.lean:294-297: def SparsePerturb (r : ℕ) (zStar : ℝ) : Prop := ⏎ 0 < zStar ∧ zStar ≤ baseZeta r ∧ ∀ G : Grid, G.b ≤ SparseScalar.B0 → G.Allowed r → ⏎ ∀ z ∈ Set.Icc 0 zStar, ∀ E : Finset G.Choices, E.Nonempty → ⏎ G.eventDistance E z < SparseScalar.slack G.size", "ComparatorChallenges/CoordinateSweeps.lean:300-303: theorem all_grid_moment {r : ℕ} (hr : ScaleLarge r) (hrS : SparseScalar.ScaleReady r) ⏎ (hS : ∀ G : Grid, G.Allowed r → ScalarInduction.MainSizeReady SparseScalar.q G.size) ⏎ {zStar : ℝ} (hζ : SparsePerturb r zStar) {z : ℝ} (hz : z∈Set.Icc 0 zStar) : ⏎ ∀ G : Grid, G.Allowed r → MomentBound G SparseScalar.q z := by", "ComparatorChallenges/CoordinateSweeps.lean:307: theorem conditional_main : ConditionalMain := by" ] } ], "readings": [ "first/batch6.json", "flags/flags4.json" ] }, { "family": "240", "title": "Shelah's eventual categoricity and the prescribed-threshold obstruction", "headline": "Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply categoricity throughout the same tail. Categoricity means uniqueness up to isomorphism at a given cardinality. Under the continuum hypothesis, a proposed specific Hanf threshold need not suffice.", "challenges": [ "CHObstruction" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply categoricity throughout the same tail", "coveredBy": [], "note": "Not stated: no challenge states the eventual-categoricity theorem (the main result of 'Eventual categoricity for abstract elementary classes'). Declared out in the scope note." }, { "clause": "Categoricity means uniqueness up to isomorphism at a given cardinality", "coveredBy": [ "CHObstruction" ], "note": "Definitional; ClassData.Categorical μ = some model of size μ exists and any two models of size μ are isomorphic." }, { "clause": "Under the continuum hypothesis, a proposed specific Hanf threshold need not suffice", "coveredBy": [ "CHObstruction" ], "note": "Assuming CH ((2:Cardinal)^ℵ₀ = ℵ₁): ∃ a countable relational language and an AEC with Löwenheim–Skolem number ℵ₀ with two non-isomorphic models of size ℶ_{ω₂} = hanf(ℵ₀) = ℶ_{(2^ℵ₀)^+}, yet categorical in every μ ≥ ℶ_{(2^ℵ₁)^+}." } ], "gap": "The headline's main claim — Shelah's eventual categoricity conjecture proved in ZFC (uniform threshold, transfer of categoricity through the tail) — is not stated; only the CH obstruction to the prescribed Hanf threshold is.", "scopeNoteOnGap": "docs/240.md:11: …The later canonical-point obstruction and the separate eventual-categoricity theorem are not included.", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "241", "title": "Rigidity of the Turing degrees", "headline": "Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem. Thus no nontrivial relabeling of degrees preserves the ordering by relative computability.", "challenges": [ "DegreeRigidity" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem", "coveredBy": [ "DegreeRigidity" ], "note": "∀ π : Degree ≃o Degree, ∀ a, π a = a, where Degree is the antisymmetrisation of ℕ → Bool under Mathlib's TuringReducible (relative partial recursiveness of the characteristic functions)." }, { "clause": "Thus no nontrivial relabeling of degrees preserves the ordering by relative computability", "coveredBy": [ "DegreeRigidity" ], "note": "Restatement of the same." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "242", "title": "Single-fold Diophantine representations and undecidability under an at-most-one-solution promise", "headline": "Every recursively enumerable set of tuples of natural numbers has a Diophantine representation with exactly one auxiliary solution for each member and none for nonmembers. This proves the single-fold conjecture and hence the finite-fold conjecture. Diophantine solvability over the nonnegative integers remains undecidable even with an at-most-one-solution promise.", "challenges": [ "SingleFold" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Every recursively enumerable set of tuples of natural numbers has a Diophantine representation with exactly one auxiliary solution for each member and none for nonmembers", "coveredBy": [ "SingleFold" ], "note": "For n >= 1 and every REPred S subset of N^n there are m >= 1 and an integer polynomial P in n + m variables with: a in S iff some w in N^m has P(a,w) = 0, and any two such witnesses are equal." }, { "clause": "This proves the single-fold conjecture and hence the finite-fold conjecture", "coveredBy": [ "SingleFold" ], "note": "Single-fold is stated; finite-fold follows trivially (one witness is finitely many)." }, { "clause": "Diophantine solvability over the nonnegative integers remains undecidable even with an at-most-one-solution promise", "coveredBy": [], "note": "Not stated. It follows from the single-fold theorem plus the existence of a non-computable r.e. set, but it is not in the challenge." } ], "gap": "The undecidability-under-promise consequence (clause 3) is not stated.", "scopeNoteOnGap": "docs/242.md:11: \"The consequence that solvability remains undecidable under an at-most-one-solution promise is not separately formalized.\"", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "243", "title": "Separating choiceless counting from polynomial time and witnessed choice", "headline": "Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽3 defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot express. A separate result shows that adding witnessed symmetric choice strictly increases expressive power. Both separations hold for the full counting formalism, allowing hereditarily finite sets of arbitrary finite rank.", "challenges": [ "ChoicelessPolynomialTime", "WitnessedChoice" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽3 defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot express.", "coveredBy": [ "ChoicelessPolynomialTime" ], "note": "main: the query (∃ λ, μ over ZMod 3 solving the normalized linear system encoded by 8 binary relations) is isomorphism-invariant, decidable in polynomial time on ordered encodings by a finite-alphabet TM2, and not EvaluationDefinable by any polynomially time- and space-bounded BGS program with cardinality." }, { "clause": "A separate result shows that adding witnessed symmetric choice strictly increases expressive power.", "coveredBy": [ "WitnessedChoice" ], "note": "∃ sentence φ with exactly one wsc occurrence, Boolean-valued on every finite input, whose true-model class differs from that of every wsc-free (CPT+C) sentence." }, { "clause": "Both separations hold for the full counting formalism, allowing hereditarily finite sets of arbitrary finite rank.", "coveredBy": [ "ChoicelessPolynomialTime", "WitnessedChoice" ], "note": "Both models use Mathlib Lists A quotients (HF sets over atoms, any finite rank) with a card operation." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": "Two large hand-built operational semantics. MATCHES assumes the ASM and logic models are faithful to BGS CPT+C (nothing found that makes them weaker). Counting intermediate objects in the space bound is an encoding choice a specialist should confirm is polynomially equivalent.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [ { "challenge": "ChoicelessPolynomialTime", "flag": "LARGE STATEMENT (561 lines): read through the closure of main: Input.query/Iso, OrdinaryPolynomialTime (TableMachine.code, GaussMachine.boolCode, Input.ordered), FullCPT.EvaluationDefinable → Program, Rule.updates, Term.eval, State/applyUpdates/Consistent, Program.run, Program.occurring → active/critical/familyClosure/closure, stepRoots → Term.roots/Rule.roots. No degenerate definition found: programs can halt and accept (the accept/halt flags are updatable 0-ary controls), so definability is not vacuously impossible." }, { "challenge": "ChoicelessPolynomialTime", "flag": "VACUITY? checked, none: the non-definability conjunct would be trivial if no program could ever accept; this one can (e.g. one that sets halt and accept at once defines the trivial query)." }, { "challenge": "WitnessedChoice", "flag": "LARGE STATEMENT (475 lines): read through the closure of MainStatement: Sentence.eval/BooleanOnAllInputs/TrueModels/isCPT, Term.eval/Formula.eval, collect, ordinaryIteration/iterationLoop, witnessedIteration/GoodWSC/FirstPath/ChoicePrefix/StrictPrefix/WitnessesHistory, permutationGraph, Automorphism, FixesParameters, the HF MulAction, map, closure/TC. No degenerate definition found." } ], "flagReadings": [ { "challenge": "ChoicelessPolynomialTime", "flag": "VACUITY? checked, none: the non-definability conjunct would be trivial if no program could ever accept; this one can (e.g. one that sets halt and accept at once defines the trivial query).", "decision": "cleared", "why": "The model is not degenerate, and it defines more than the constant queries the first reader exhibited. Take a one-step program with body parallel(update halt := ordinal 1, update accept := not(equal(comprehend x ∈ atoms | input Ed x x . x, ordinal 0))). From the initial store, halt reads ∅ ≠ {∅}, so the step fires. The two updates hit different locations, so State.Consistent holds. run 1 halts and accepts iff some a has Ed a a, a non-constant, isomorphism-invariant query. It runs in time 1, and its space is linear in |A|: the occurring set is the hereditary closure of the atoms, allAtoms, ∅, {∅} and one subset of the atoms. So EvaluationDefinable is a non-trivial class, and ¬ EvaluationDefinable(query) is a real separation, not a vacuous one. Lane K has not run ChoicelessPolynomialTime.", "evidence": [ "lean/ComparatorChallenges/ChoicelessPolynomialTime.lean:344: | .input r a b, v => Operational.boolean (Operational.inputRelation rel r (eval rel s a v) (eval rel s b v))", "lean/ComparatorChallenges/ChoicelessPolynomialTime.lean:349: | .comprehend n bound guard body, v =>", "lean/ComparatorChallenges/ChoicelessPolynomialTime.lean:370: | .parallel a b, v => updates rel s a v ∪ updates rel s b v", "lean/ComparatorChallenges/ChoicelessPolynomialTime.lean:397: if State.Consistent u then some (s.applyUpdates u) else none", "lean/ComparatorChallenges/ChoicelessPolynomialTime.lean:468: ∃ h st, h ≤ time.eval (Fintype.card A) ∧ P.run rel h = some st ∧ P.flag st .halt ∧ P.flag st .accept ∧", "lean/ComparatorChallenges/ChoicelessPolynomialTime.lean:553: ¬ FullCPT.EvaluationDefinable (fun {_} [_] I => I.query) := by" ] } ], "readings": [ "first/batch5.json", "flags/flags1.json" ] }, { "family": "244", "title": "The Partition Principle does not imply Choice", "headline": "Assuming ZF is consistent, constructs a model in which every surjective image of a set injects into that set, yet the axiom of choice fails. Choice for ordinal-indexed families still holds. From any countable transitive model of ZFC, a separate construction gives a transitive symmetric extension with these properties and no new countable sequences of ground-model elements.", "challenges": [ "PartitionPrinciple", "PartitionConsistency" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Assuming ZF is consistent, constructs a model in which every surjective image of a set injects into that set, yet the axiom of choice fails", "coveredBy": [ "PartitionConsistency" ], "note": "Consistent ZF -> Consistent (ZF + PP + ACwo + not AC), with syntactic consistency in an in-file natural-deduction system." }, { "clause": "Choice for ordinal-indexed families still holds", "coveredBy": [ "PartitionConsistency" ], "note": "ACwo is in targetTheory." }, { "clause": "From any countable transitive model of ZFC, a separate construction gives a transitive symmetric extension with these properties and no new countable sequences of ground-model elements", "coveredBy": [ "PartitionPrinciple" ], "note": "Partial. The ground must also contain an internal strongly inaccessible K (extra hypothesis). The conclusion is only some transitive N containing M with N satisfying ZF + PP + not AC. Not stated: ACwo in N, that N is a symmetric extension, no new countable sequences of ground elements (nor the abstract's \"same ordinals\")." } ], "gap": "In the countable-transitive-model clause, the Lean adds an internal strongly inaccessible cardinal to the hypotheses and drops from the conclusion ACwo-in-N, \"symmetric extension\" and \"no new countable sequences of ground-model elements\". The relative-consistency clauses are fully stated.", "scopeNoteOnGap": "docs/244.md:11: \"The linked model-construction result also constructs a transitive model of the Partition Principle without Choice from an externally countable transitive ground model satisfying the stated axioms, including Choice, and containing an internal strongly inaccessible cardinal. The paper's stronger transitive-model preservation assertions are outside that selected construction.\" (ACwo-in-N and \"symmetric\" are declared only by that positive description.)", "note": null, "firstSuspicious": "The two challenges use opposite variable conventions (levels vs indices). I checked every formula under its own convention; an independent re-check of PartitionConsistency.replacement (lines 280-294) is cheap insurance.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "PartitionPrinciple", "flag": "NARROWER (declared): the extra internal-inaccessible hypothesis (hK, standard) and the weaker conclusion (no ACwo in N, no symmetric-extension or countable-sequence clause). Lean: lines 269-272. Scope note: docs/244.md:11." }, { "challenge": "PartitionPrinciple", "flag": "VACUITY? (low): the hypotheses need a countable transitive model of ZFC with an internal inaccessible inside ZFSet.{u}. That is satisfiable when the ambient universe has enough strength (e.g. two inaccessibles); it is not vacuous in general." } ], "flagReadings": [ { "challenge": "PartitionPrinciple", "flag": "VACUITY? (low): the hypotheses need a countable transitive model of ZFC with an internal inaccessible inside ZFSet.{u}. That is satisfiable when the ambient universe has enough strength (e.g. two inaccessibles); it is not vacuous in general.", "decision": "cleared", "why": "The hypotheses require a countable transitive M ⊆ ZFSet.{u} satisfying pairing, union, power set, separation, collection, infinity and choice, together with K ∈ M that is inaccessible in M. I re-read the de Bruijn encodings of the hypothesis axioms: choice (196-198) through choiceFunctionFormula (85-88) and orderedPairFormula (67-70, Kuratowski pairs matching ZFSet.pair), infinity (192-194) and powerSet (169-172). All read as the standard axioms, so no hypothesis is accidentally contradictory. GroundStandardInaccessible (142-144) reads as 'uncountable regular strong-limit cardinal of M', with groundPower κ = P(κ) ∩ M = P^M(κ). Such an M exists in Lean's intended semantics (ZFC + ω inaccessibles). With inaccessibles κ₀ < κ₁, a countable elementary submodel of V_{κ₁} collapses to a ctm of ZFC + 'κ₀ is inaccessible'. That ctm is hereditarily countable, so it lies in the interpretation of ZFSet.{u}. By soundness Lean cannot refute the hypotheses, so the theorem is not vacuously provable. The extra internal inaccessible is a narrowing, and docs/244.md:11 declares it. It is not a vacuity.", "evidence": [ "lean/ComparatorChallenges/PartitionPrinciple.lean:86: .conj (.allMem f (exMem a.succ (exMem 0 (orderedPairFormula 2 1 0))))", "lean/ComparatorChallenges/PartitionPrinciple.lean:87: (.allMem a (exMem 0 (.conj (pairMembershipFormula 1 0 f.succ.succ)", "lean/ComparatorChallenges/PartitionPrinciple.lean:88: (.allMem 1 (imp (pairMembershipFormula 2 0 f.succ.succ.succ) (.equal 0 1))))))", "lean/ComparatorChallenges/PartitionPrinciple.lean:143: GroundCardinal M K ∧ ¬ M.groundEmbeddable K ZFSet.omega ∧", "lean/ComparatorChallenges/PartitionPrinciple.lean:144: GroundCofinalRegular M K ∧ GroundCardinalStrongLimit M K", "lean/ComparatorChallenges/PartitionPrinciple.lean:197: .all (imp (BoundedFormula.allMem 0 (BoundedFormula.nonemptyFormula 0)).toFormula", "lean/ComparatorChallenges/PartitionPrinciple.lean:198: (ex (BoundedFormula.choiceFunctionFormula 1 0).toFormula))", "lean/ComparatorChallenges/PartitionPrinciple.lean:269: {K : ZFSet.{u}} (hK : K ∈ M.sets) (standard : GroundStandardInaccessible M K) :", "lean/docs/244.md:11: The linked model-construction result also constructs a transitive model of the Partition Principle without Choice from an externally countable transitive ground model satisfying the stated axioms, including Choice, and containing an internal strongly inaccessible cardinal. The paper's stronger transitive-model preservation assertions are outside that selected construction." ] } ], "readings": [ "first/batch7.json", "flags/flags2.json" ] }, { "family": "245", "title": "Weak normalization implies strong normalization in pure type systems", "headline": "Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a β-normal form, every β-reduction sequence terminates. This resolves the β-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts.", "challenges": [ "TypeSystemNormalization" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a β-normal form, every β-reduction sequence terminates", "coveredBy": [ "TypeSystemNormalization" ], "note": "For every specification P (sorts in any Type u, arbitrary axiom and rule relations): if every M legal in every valid context Γ (∃ A, Γ ⊢ M : A ∨ Γ ⊢ A : M) reaches a β-normal form, then every such M is accessible for β-reduction (no infinite reduction sequence)." }, { "clause": "This resolves the β-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts", "coveredBy": [ "TypeSystemNormalization" ], "note": "Axioms and rules are arbitrary relations (no functionality); contexts are arbitrary valid contexts; β reduces inside λ/Π annotations." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "246", "title": "Cannon's conjecture", "headline": "Every word-hyperbolic group with boundary homeomorphic to S2 admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture. Every torsion-free such group is therefore the fundamental group of a closed hyperbolic three-manifold.", "challenges": [ "CannonGeometricAction" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Every word-hyperbolic group with boundary homeomorphic to S2 admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture.", "coveredBy": [ "CannonGeometricAction" ], "note": "Stated directly." }, { "clause": "Every torsion-free such group is therefore the fundamental group of a closed hyperbolic three-manifold.", "coveredBy": [ "CannonGeometricAction" ], "note": "Implied by a standard step, not stated: in a torsion-free group the finite kernel and finite stabilizers are trivial, so the action is free and properly discontinuous and H³/G is a closed hyperbolic 3-manifold with π₁ = G." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch1.json" ] }, { "family": "247", "title": "An infinite finitely presented residually finite 2-group and a finitely presented nil algebra", "headline": "Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also yields an infinite-dimensional finitely presented nil associative 𝔽2-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions.", "challenges": [ "PeriodicGroup" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "answering the finitely presented Burnside problem negatively", "status": "stated", "coveredBy": [ "PeriodicGroup" ], "evidence": [ "lean/ComparatorChallenges/PeriodicGroup.lean:45: ∃ (G : Type) (_ : Group G), Infinite G ∧ Group.IsFinitelyPresented G ∧ Periodic G", "lean/ComparatorChallenges/PeriodicGroup.lean:41: Infinite (Steinberg 12 R) ∧ Group.IsFinitelyPresented (Steinberg 12 R) ∧", "lean/ComparatorChallenges/PeriodicGroup.lean:51: theorem thm_main : MainStatement ∧ BurnsideStatement := by" ], "note": "An infinite, finitely presented, periodic group exists, realized as St₁₂(R) for a ring R that is a ZMod 2-algebra. Periodic means every element has some finite order. Not trivially witnessed: St₁₂(𝔽₂) is finite and St₁₂(𝔽₂[t]) is not periodic." }, { "clause": "Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order", "status": "unlinked", "coveredBy": [ "PeriodicGroup" ], "manuscript": "An-infinite-finitely-presented-residually-finite-2-group-October-5-2026", "evidence": [ "lean/docs/247.md:5: - [An infinite finitely presented periodic group](../../preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/paper.pdf)", "lean/ComparatorChallenges/PeriodicGroup.lean:37: def Periodic (G : Type u) [Group G] : Prop := ∀ g : G, ∃ m : ℕ, 0 < m ∧ g ^ m = 1", "CONTENTS.md:6119: is infinite, ordinarily finitely presented, and residually finite, and every element of G has finite 2-power order." ], "note": "Infinite + finitely presented + periodic is stated; 'residually finite' and 'finite 2-power order' (and so 'even in this class') are not. Both come from the October-5 manuscript, which the note does not link. The linked September-23 paper claims neither (its build has no 'residually finite' or '2-power'). docs/247.md:9 says only 'no common exponent is asserted', which names a different omission." }, { "clause": "The construction also yields an infinite-dimensional finitely presented nil associative 𝔽2-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions.", "status": "declared", "coveredBy": [], "manuscript": "An-infinite-finitely-presented-periodic-group-September-23-2026", "evidence": [ "lean/docs/247.md:9: The paper's separate nil-algebra and radical-algebra conclusions are outside these selected statements." ], "note": "The algebraic unitization (Kurosh) is the nil algebra's unitization, and its algebraicity is a consequence of nilness, so 'nil-algebra ... conclusions' covers it." } ], "gap": "Residual finiteness and the 2-power order of every element (the October 5 manuscript's contribution; undeclared), plus the nil algebra and algebraic-unitization clause (declared outside).", "scopeNoteOnGap": "partly declared: lean/docs/247.md:9: \"The paper's separate nil-algebra and radical-algebra conclusions are outside these selected statements.\" and lean/docs/247.md:9: \"Periodicity means that each element has some finite order; no common exponent is asserted.\"; silent on residual finiteness and on 2-power orders, and links only the September 23 manuscript (247.md:5), not the October 5 'An infinite finitely presented residually finite 2-group'", "note": "The negative Burnside answer is stated, and the nil-algebra/Kurosh clause is declared out. Residual finiteness and 2-power orders come from the unlinked October-5 manuscript, so the family is UNLINKED. The first reader had the same gap as UNDECLARED, before amendment 3.", "firstSuspicious": "Same pattern as 084, 186, 197.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch6.json", "second/second2.json" ] }, { "family": "248", "title": "Thompson's group F is nonamenable", "headline": "Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem.", "challenges": [ "ThompsonNonamenability" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable", "coveredBy": [ "ThompsonNonamenability" ], "note": "F = increasing homeomorphisms of [0,1], piecewise affine with dyadic breakpoints and slopes 2^k; no positive normalised left-invariant mean on ℓ^∞(F)." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "249", "title": "A finitely generated Eilenberg–Ganea counterexample", "headline": "Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture. It has no two-dimensional classifying space, even with infinitely many cells.", "challenges": [ "EilenbergGanea" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture", "coveredBy": [ "EilenbergGanea" ], "note": "A specific group (the kernel of the height map on a right-angled Artin group) with Group.FG, ResiduallyFinite, cd = 2 (projective dimension of the trivial Z[G]-module), a 3-dimensional classifying CW space, and no 2-dimensional one." }, { "clause": "It has no two-dimensional classifying space, even with infinitely many cells", "coveredBy": [ "EilenbergGanea" ], "note": "not HasClassifyingSpace.{u} SourceGroup 2 for every universe u, with no finiteness condition on the cells." } ], "gap": null, "scopeNoteOnGap": "no gap", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "250", "title": "Boone–Higman embeddings with higher finiteness", "headline": "A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture. The target can have type F∞: a classifying space with finitely many cells in each dimension. A single group of type F∞ can also contain every finitely presented group.", "challenges": [ "BooneHigman", "SimpleOvergroups", "UniversalFInfinity" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture.", "coveredBy": [ "BooneHigman" ], "note": "For finitely generated G: (∃ finite generating tuple with computable word problem) ↔ ∃ finitely presented simple H (same universe) with an injective homomorphism G → H." }, { "clause": "The target can have type F∞: a classifying space with finitely many cells in each dimension.", "coveredBy": [ "SimpleOvergroups" ], "note": "f.g. G with decidable word problem embeds in a nontrivial simple H of type F∞ (a K(H,1) CW complex with finitely many cells in each dimension); F∞ implies finitely presented, so the target is a finitely presented simple group of type F∞." }, { "clause": "A single group of type F∞ can also contain every finitely presented group.", "coveredBy": [ "UniversalFInfinity" ], "note": "∃ H of type F∞ ∀ finitely presented G (same universe) ∃ injective G → H; H is chosen before G." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch3.json" ] }, { "family": "251", "title": "Amenability, unitarizability, and strong Ulam stability", "headline": "Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability is also equivalent to strong Ulam stability: sufficiently accurate unitary approximate representations are uniformly close in operator norm to genuine representations on the same, possibly infinite-dimensional, Hilbert space.", "challenges": [ "DixmierAllDiscrete", "Dixmier" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation.", "status": "stated", "coveredBy": [ "DixmierAllDiscrete", "Dixmier" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/DixmierAllDiscrete.lean:36: (Amenable G ↔ Unitarizable.{u, max u v} G) ∧" ] }, { "clause": "For countable discrete groups, amenability is also equivalent to strong Ulam stability: sufficiently accurate unitary approximate representations are uniformly close in operator norm to genuine representations on the same, possibly infinite-dimensional, Hilbert space.", "status": "unlinked", "coveredBy": [], "manuscript": "Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026", "evidence": [ "preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026/build/strong-ulam-stability.tex:43: A countable discrete group is amenable if and only if it is strongly Ulam", "lean/docs/251.md:5: - [Unitarizability implies amenability for discrete groups]", "lean/docs/251.md:9: The formalization establishes this equivalence for every discrete group.", "lean/docs/251.md:1: # Amenability, unitarizability, and strong Ulam stability" ] } ], "gap": "the strong-Ulam-stability characterization of amenability for countable groups", "scopeNoteOnGap": "silent — docs/251.md links only the unitarizability manuscript and describes only it (docs/251.md:9: \"The formalization establishes this equivalence for every discrete group.\"); the October-5 'Strong-Ulam-Stability-Characterizes-Amenability' manuscript is neither linked nor mentioned", "note": "The Dixmier equivalence is stated for every discrete group (invariant mean on bounded functions; similar to unitary = an invertible S making every S π(g) S⁻¹ isometric; encoding: Hilbert spaces in universe max u v). No approximate representation appears in either challenge; strong Ulam stability is the October 5 manuscript, which the note does not link (only its heading names it). NARROWER — UNLINKED; I differ from the first reader by the amendment-3 word only.", "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch8.json", "second/second3.json" ] }, { "family": "252", "title": "A torsion-free hyperbolic group that is neither residually finite nor linear over any field", "headline": "Constructs a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively. One fixed nonidentity element is killed by every finite-dimensional linear representation over every commutative field, so the group is not linear over any such field.", "challenges": [ "TorsionFreeHyperbolic" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively", "coveredBy": [ "TorsionFreeHyperbolic" ], "note": "∃ a group G (in Type) that is torsion-free, word-hyperbolic (a finite S whose Cayley graph is connected and whose geodesic triangles are δ-thin at the vertex level) and not Group.ResiduallyFinite." }, { "clause": "One fixed nonidentity element is killed by every finite-dimensional linear representation over every commutative field, so the group is not linear over any such field", "coveredBy": [], "note": "Not stated. It follows from the stated conjuncts by Malcev's theorem (finitely generated linear groups are residually finite; any element of the finite residual is killed by every finite-dimensional representation), but that classical theorem is not part of the challenge. Declared in the scope note." } ], "gap": "The non-linearity clause (one fixed nonidentity element killed by every finite-dimensional representation over every field) is not stated; it is a classical consequence (Malcev) of the stated non-residual-finiteness of a finitely generated group, not a separate conclusion.", "scopeNoteOnGap": "docs/252.md:11: Nonlinearity over every field is not a separate conclusion of the statement linked below.", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "253", "title": "An infinite finitely presented simple amenable group", "headline": "Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously.", "challenges": [ "SimpleAmenable" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously.", "coveredBy": [ "SimpleAmenable" ], "note": "∃ group G (Type 0): Infinite, Group.IsFinitelyPresented, IsSimpleGroup, Følner-amenable." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch5.json" ] }, { "family": "254", "title": "Classifying spaces and geometric obstructions for Artin groups", "headline": "The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin $`K(\\pi,1)`$ conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture. An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT$`(0)`$ space.", "challenges": [ "HarmonicArtin", "ArtinCAT0", "ArtinParabolicIntersections" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin $`K(\\pi,1)`$ conjecture.", "coveredBy": [ "HarmonicArtin" ], "note": "Stated as contractibility of the universal cover, encoded combinatorially; the scope note calls this the equivalent statement." }, { "clause": "Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture.", "coveredBy": [ "ArtinParabolicIntersections" ], "note": "unconditional_arbitrary_intersections: sInf of any set of parabolics is parabolic (and equals an intersection of ≤ rank members)." }, { "clause": "An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT$`(0)`$ space.", "coveredBy": [ "ArtinCAT0" ], "note": "Explicit 116-generator matrix with labels 1 (diagonal), 2, 3, ∞; no geometric action on any nonempty proper CAT(0) space." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": "HarmonicArtin's conclusion rests on a combinatorial encoding of the Salvetti universal cover (nerve of the lifted face preorder); a second reader familiar with Salvetti complexes should confirm the face relation (minimal coset representatives, positive lifts) matches the CW structure.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch1.json" ] }, { "family": "255", "title": "Quasi-isometric recognition of virtually polycyclic groups", "headline": "Proves that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, resolving the Eskin–Fisher–Whyte lattice-recognition conjecture. Equivalently, a group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a uniform lattice in some such Lie group, possibly a different one.", "challenges": [ "PolycyclicRecognition" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, resolving the Eskin–Fisher–Whyte lattice-recognition conjecture", "coveredBy": [ "PolycyclicRecognition" ], "note": "group_recognition: for finitely generated P, J and a coarse equivalence P → J (bornologous maps in both directions with finite error sets — for finitely generated groups this is quasi-isometry of word metrics), P virtually polycyclic ⇒ J virtually polycyclic, and some finite-index Γ ≤ J is isomorphic to a uniform lattice in a simply connected solvable Lie group." }, { "clause": "Equivalently, a group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a uniform lattice in some such Lie group, possibly a different one", "coveredBy": [ "PolycyclicRecognition" ], "note": "lattice_recognition: for a finite-covolume lattice Λ (discrete, G-invariant Borel probability on G/Λ) in a simply connected solvable Lie group S and a finitely generated J coarsely equivalent to Λ, some finite-index Γ ≤ J is isomorphic to a uniform lattice Δ in some (possibly different) simply connected solvable Lie group R." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "256", "title": "Nonsingular systems of equations over arbitrary groups", "headline": "Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group.", "challenges": [ "Kervaire" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture.", "status": "unlinked", "coveredBy": [], "manuscript": "Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026", "evidence": [ "CONTENTS.md:6317: Every finite nonsingular system of equations over an arbitrary group has a simultaneous solution in an overgroup. This proves Howie's conjecture on nonsingular systems.", "lean/docs/256.md:5: - [The Kervaire theorem for groups](../../preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf)", "lean/ComparatorChallenges/Kervaire.lean:17: exponentSum w = 1 ∨ exponentSum w = -1" ], "why": "Only one equation in one unknown with exponent sum ±1 is stated; even the one-equation nonsingular case (exponent sum ≠ 0) is not." }, { "clause": "The coefficient group embeds in the presented quotient.", "status": "unlinked", "coveredBy": [], "manuscript": "Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026", "evidence": [ "CONTENTS.md:6317: Every finite nonsingular system of equations over an arbitrary group has a simultaneous solution in an overgroup.", "lean/ComparatorChallenges/Kervaire.lean:32: (hw : Unimodular w) : Function.Injective (coefficientMap w) := by" ], "why": "In the headline this sentence is the conclusion of the nonsingular-systems theorem (the Howie paper's). Only its single-relator, exponent-sum ±1 instance is stated (Kervaire.coefficient_injective, the linked paper's main theorem)." }, { "clause": "A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group.", "status": "stated", "coveredBy": [ "Kervaire" ], "evidence": [ "lean/ComparatorChallenges/Kervaire.lean:32: (hw : Unimodular w) : Function.Injective (coefficientMap w) := by", "preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/build/sections/introduction.tex:43: For every other integer $d=p(w)$, the map $p$ induces a surjection from", "preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/build/sections/introduction.tex:44: the quotient onto the nontrivial group $\\Z/d\\Z$, where $\\Z/0\\Z=\\Z$." ], "encoding": [ "adjoining one generator and one relation = (A ∗ ℤ) ⧸ normalClosure {w} (Monoid.Coprod A (Multiplicative ℤ)); A in any universe" ], "why": "Implied by an elementary step, not a classical theorem: if the exponent sum d of w is ±1, injectivity embeds A ≠ 1 in the quotient; otherwise the exponent-sum map induces a surjection of the quotient onto ℤ/d (ℤ for d = 0), which is nontrivial. This is the paper's own two-line corollary." } ], "gap": "Howie's theorem on finite nonsingular systems (the headline's lead clause, from the October 5 manuscript) is not stated, not even the general one-equation case (exponent sum != 0). The coefficient-embedding clause is stated only for one unimodular relator.", "scopeNoteOnGap": "silent. docs/256.md links only the Kervaire paper (line 5) and describes only the unimodular injectivity (line 9: \"The linked formalization proves the stronger coefficient-injectivity statement used in the paper: ...\"). It never says that the nonsingular-systems (Howie) theorem is left out.", "note": "Kervaire's conjecture is stated (via the stronger unimodular coefficient injectivity plus the exponent-sum step); Howie's nonsingular-systems theorem and its embedding conclusion come only from the October 5 paper, which the note does not link. So NARROWER — UNLINKED. I read the clauses as the first reader did; the word differs only because UNLINKED did not exist for them.", "firstSuspicious": "The scope note's title is the family title \"Nonsingular systems of equations over arbitrary groups\", but its body covers only Kervaire. A reader might take the title as a declaration; I did not.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "Kervaire", "flag": "NARROWER — UNDECLARED: one unimodular equation instead of finite nonsingular systems. Lean: lines 16-17, 31-32. Headline: the first clause. Scope note: silent (docs/256.md:5, 9)." } ], "flagReadings": null, "readings": [ "first/batch7.json", "second/second4.json" ] }, { "family": "257", "title": "A hyperbolic group without a geometric CAT(0) action", "headline": "Answers negatively whether every word-hyperbolic group is a CAT(0) group. Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete $`\\mathop{\\mathrm{CAT}}\\nolimits (0)`$ space, in any dimension.", "challenges": [ "HyperbolicObstruction" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Answers negatively whether every word-hyperbolic group is a CAT(0) group.", "coveredBy": [ "HyperbolicObstruction" ], "note": "The witness group is word-hyperbolic and has no geometric action on a proper complete CAT(0) space." }, { "clause": "Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete $`\\mathop{\\mathrm{CAT}}\\nolimits (0)`$ space, in any dimension.", "coveredBy": [ "HyperbolicObstruction" ], "note": "Finite connected aspherical simplicial complex K, G = π₁(K); X ranges over every nonempty proper complete CAT(0) metric space in universe u." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "260", "title": "Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions", "headline": "Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension n ≥ 3, under dominant energy, weak future trapping, positive enclosing area, and the stated decay assumptions. It bounds invariant ADM mass below using minimum enclosing area, with equality rigidity under additional horizon hypotheses. Charged upper-area bounds treat dyonic three-dimensional data; the higher-dimensional purely electric extension uses the matched neutral theorem.", "challenges": [ "CKSBondiPenrose" ], "word": "DIFFERENT", "firstWord": "DIFFERENT", "secondWord": "DIFFERENT", "disagreement": null, "clauses": [ { "clause": "Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension n ≥ 3, under dominant energy, weak future trapping, positive enclosing area, and the stated decay assumptions.", "status": "unlinked", "coveredBy": [], "manuscript": "The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026", "evidence": [ "CONTENTS.md:6475: We prove the sharp spacetime Penrose inequality for smooth one-ended initial-data exteriors in every spatial dimension n ≥ 3, under the dominant energy condition, weak future trapping, the stated differentiated decay and positive enclosing area.", "lean/ComparatorChallenges/CKSBondiPenrose.lean:3312: theorem area_controlled_end_replacement", "lean/ComparatorChallenges/CKSBondiPenrose.lean:3591: theorem schwarzschild_equality_examples (m : ℝ) (hm : 0 < m) :", "lean/docs/260.md:13: The implementation retains a conditional inequality for the connected marginal-boundary case, assuming the asymptotically flat exterior Penrose inequality. This supporting result is not selected as a substitute for the paper's main inequality.", "lean/docs/260.md:5: - [Area-controlled end replacement and the Bondi–Penrose inequality in the CKS class](../../preprints/Area-controlled-end-replacement-and-the-Bondi-Penrose-inequality-in-the-CKS-class-September-27-2026/paper.pdf)" ] }, { "clause": "It bounds invariant ADM mass below using minimum enclosing area, with equality rigidity under additional horizon hypotheses.", "status": "unlinked", "coveredBy": [], "manuscript": "The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026; Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026", "evidence": [ "CONTENTS.md:6493: Equality in the spacetime Penrose inequality identifies the original initial data as a spacelike slice of a Schwarzschild–Tangherlini exterior, smoothly attached to its horizon, under the stated outermostness and decay hypotheses.", "lean/ComparatorChallenges/CKSBondiPenrose.lean:3575: OAI.CKSMain.bondiMass d = m ∧", "lean/docs/260.md:5: - [Area-controlled end replacement and the Bondi–Penrose inequality in the CKS class](../../preprints/Area-controlled-end-replacement-and-the-Bondi-Penrose-inequality-in-the-CKS-class-September-27-2026/paper.pdf)" ] }, { "clause": "Charged upper-area bounds treat dyonic three-dimensional data", "status": "unlinked", "coveredBy": [], "manuscript": "The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026", "evidence": [ "CONTENTS.md:6399: we prove the sharp charged spacetime Penrose upper-area inequality for one-ended three-dimensional initial data with source-free electric and magnetic fields.", "lean/docs/260.md:5: - [Area-controlled end replacement and the Bondi–Penrose inequality in the CKS class](../../preprints/Area-controlled-end-replacement-and-the-Bondi-Penrose-inequality-in-the-CKS-class-September-27-2026/paper.pdf)" ] }, { "clause": "the higher-dimensional purely electric extension uses the matched neutral theorem.", "status": "unlinked", "coveredBy": [], "manuscript": "A-Charged-Reduction-of-the-Spacetime-Penrose-Inequality-in-Spatial-Dimensions-at-Least-Four-October-5-2026", "evidence": [ "CONTENTS.md:6408: Using the companion neutral spacetime Penrose theorem exactly at its stated strong-decay, future-trapped, positive-area, and future-timelike scope, we prove the sharp purely electric upper-area inequality in every spatial dimension n ≥ 4", "lean/docs/260.md:5: - [Area-controlled end replacement and the Bondi–Penrose inequality in the CKS class](../../preprints/Area-controlled-end-replacement-and-the-Bondi-Penrose-inequality-in-the-CKS-class-September-27-2026/paper.pdf)" ] } ], "gap": "The entire headline. No Penrose inequality of any kind is stated (neither the headline's ADM / asymptotically-flat enclosing-area inequality in any dimension n >= 3, nor even the CKS paper's Bondi-mass inequality), no equality rigidity, no charged dyonic three-dimensional upper-area bound, no higher-dimensional purely electric extension. What is stated is a three-dimensional CKS-class end-replacement construction (from one of the family's 13 manuscripts) and Schwarzschild sharpness examples for an inequality that is itself not stated. The union neither implies the headline nor is implied by it.", "scopeNoteOnGap": "docs/260.md:11: The Comparator also includes canonical Schwarzschild equality examples at every positive mass, with $m_{\\mathrm{Bondi}}=m$ and enclosing area $16\\pi m^2$. The paper's general Bondi–Penrose inequality for possibly disconnected weakly trapped boundaries remains unformalized. | docs/260.md:13: The implementation retains a conditional inequality for the connected marginal-boundary case, assuming the asymptotically flat exterior Penrose inequality. This supporting result is not selected as a substitute for the paper's main inequality. | (Otherwise silent: docs/260.md:5 links only the CKS paper, and the note never names the headline's asymptotically flat enclosing-area theorem in n >= 3, its rigidity, the charged dyonic bounds or the higher-dimensional electric extension; line 13 implies the asymptotically flat Penrose inequality is assumed, not formalized.)", "note": "No headline clause is stated: the only challenge states the CKS end replacement (hyperboloidal end → asymptotically flat ends with ADM mass → Bondi mass, area loss → 0) and Schwarzschild Bondi-equality examples, which neither imply nor are implied by the asymptotically flat Penrose inequality, its rigidity, or the charged bounds — DIFFERENT. Every headline clause comes from a manuscript docs/260.md does not link (it links only the CKS paper of 13; 260.md:13 mentions the AF Penrose inequality only as the hypothesis of an unselected conditional result, which I do not read as declaring it). Flag: LARGE STATEMENT (3,596 lines); the closure of the two theorem statements (MainHypotheses, SchwarzschildExample, CKSData) carries no Penrose inequality. Agrees with the first reader.", "firstSuspicious": "(a) Word: DIFFERENT is chosen by the pre-registered definition (neither implies nor is implied by the headline): the end-replacement construction is a bridge from the CKS class into the headline theorem's input class, not a special case of the headline, and the Schwarzschild examples concern Bondi mass of hyperboloidal slices, not ADM mass. A second reader may weigh NARROWER — DECLARED on the view that the scope note honestly describes what is formalized and declares the CKS paper's main inequality unformalized (docs/260.md:11); but the scope note never says the headline's own lead theorem (AF, n >= 3, ADM) is unformalized except by implication (docs/260.md:13). (b) Areas and constraint integrability rest on measures chosen by Classical.epsilon (lines 1530, 1930); genuine given the standard existence of Riemannian volume. (c) Census marks mainResultInCatalogue = false, support = false: OpenAI does not label this challenge a supporting result even though its selected theorems are the paper's construction lemma and sharpness examples rather than its main inequality.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "CKSBondiPenrose", "flag": "LARGE STATEMENT: 3,596 lines, two theorem holes (sorry at lines 3353 and 3593), no axiom declarations, definition_names = []. Read whole. The definitional closure of the two theorems is essentially the entire file (a crude name-reference pass misses dot-notation and instance uses; manual tracing reaches every block). The bulk is proof-carrying structure — the Schwarzschild exterior's charts and smooth-manifold instance (~1117–1256), the boundary-surface charts (~1562–1823), the induced metric on cuts (~2541–2690, 2753–2898) — well-formedness proofs that cannot carry either conclusion." }, { "challenge": "CKSBondiPenrose", "flag": "VACUITY?: low. Theorem (1)'s hypotheses are jointly satisfiable if theorem (2) is proved: MainHypotheses (3541–3548) repeats exactly hK, hKsym, hOrient, hScompact, hSnonempty, hcomplete, hDEC, ht, and SchwarzschildExample asserts an inhabitant on the Schwarzschild exterior for every m > 0 (Type 0, with T2/second-countable/connected instances). Minor: ScaledComponentBound divides by (y 0)^q (line 342), so a patch with radius <= 0 would force impossible bounds at y0 <= 0 — this only strengthens the hypothesis." }, { "challenge": "CKSBondiPenrose", "flag": "JUNK?: cutArea = (area g D).toReal (line 3073) maps an infinite area to 0, and minimumEnclosingArea is a real infimum (line 3078) — neutralized: theorem (1) asserts area g D < ⊤ and area gR D < ⊤ for every D (3351) and minEnclosingArea g < ⊤ (3326); the index type is nonempty (N itself is an OuterDomain) and areas are >= 0, so the real infimum is the genuine one. In theorem (2) '= 16πm^2' is a strengthening, not a weakening." }, { "challenge": "CKSBondiPenrose", "flag": "JUNK?: riemannianVolume := Classical.epsilon (IsVolume g) (lines 1530, 1930): cut areas and constraint integrability are taken against an epsilon-chosen measure; genuine if and only if a measure with the prescribed chart densities exists (the standard Riemannian volume does). If none existed the value would be unspecified, which could not make either conclusion easier to prove. Not exploitable; noted for a second reader." }, { "challenge": "CKSBondiPenrose", "flag": "JUNK?: no effective zero-defaults in the Bochner integrals: bondiCharge integrates a smooth mass aspect (aspect_smooth, 3035) over the finite sphere measure; spatialADMEnergy/Momentum integrate continuous functions for large r (Symbol 1 includes C^∞ near infinity); the ADM momentum is identically 0 for r > 2R^2 because k = 0 there (3343) — genuine, not a default. Real.sqrt(E^2 - |P|^2) is genuine under ht (3321) / timelike (3548)." }, { "challenge": "CKSBondiPenrose", "flag": "QUANTIFIER?: matches the manuscript theorem: ∃ d' R0 eps (eps -> 0, eps uniform over all cuts D) ∀ R >= R0 ∃ gR kR G k eta with eta <= 2R^(-1/2). In theorem (2) the CKS chart d is existential, so bondiMass d = m holds for some admissible chart; chart-independence of the Bondi mass is not itself stated (acceptable for an example)." }, { "challenge": "CKSBondiPenrose", "flag": "DEFINITION HOLE: none (definition_names = [])." }, { "challenge": "CKSBondiPenrose", "flag": "NO PENROSE INEQUALITY: no inequality between any mass (ADM or Bondi) and any area occurs in the file. MainHypotheses (3538–3552: connected marginal boundary, positive area, no additional horizons) is the hypothesis bundle of the unselected conditional inequality the scope note mentions (docs/260.md:13); it is used only inside SchwarzschildExample, so the challenge certifies that Schwarzschild meets those hypotheses and attains equality without stating the inequality." } ], "flagReadings": null, "readings": [ "first/batch7.json", "second/second5.json" ] }, { "family": "261", "title": "Localization and delocalization in the Anderson model", "headline": "Resolves the predicted spectral contrast for the lattice Anderson model with independent uniform site potentials. In dimension two, every positive disorder strength gives almost surely pure-point spectrum. In every fixed dimension d ≥ 3, sufficiently weak positive disorder gives purely absolutely continuous spectrum on a fixed open interval with nonzero spectral weight.", "challenges": [ "PlanarAndersonSpectrum" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "DIFFERENT", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "DIFFERENT", "kind": "the reading" }, "clauses": [ { "clause": "Resolves the predicted spectral contrast for the lattice Anderson model with independent uniform site potentials.", "status": "unlinked", "coveredBy": [], "manuscript": "Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026 ; Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026", "partial": "a summary of clauses 2 and 3; its d ≥ 3 half is unlinked, its d = 2 half declared", "evidence": [ "preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/build/paper.tex:41-44: For each fixed dimension $d\\ge3$ and each fixed sufficiently small positive ⏎ disorder strength, we prove that the Anderson operator on $\\mathbb Z^d$ with ⏎ independent uniform site potentials almost surely has purely absolutely ⏎ continuous spectrum with nonzero weight on an open energy interval. The", "lean/docs/261.md:5: - [Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder](../../preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf)", "lean/docs/261.md:11: This selected statement identifies the spectral set. It does not assert the pure-point spectral type claimed in the accompanying paper." ] }, { "clause": "In dimension two, every positive disorder strength gives almost surely pure-point spectrum.", "status": "declared", "coveredBy": [], "manuscript": "Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026", "evidence": [ "lean/docs/261.md:11: This selected statement identifies the spectral set. It does not assert the pure-point spectral type claimed in the accompanying paper.", "lean/ComparatorChallenges/PlanarAndersonSpectrum.lean:26-28: theorem anderson_spectrum_ae {h : ℝ} (hh : 0 < h) : ⏎ ∀ᵐ v ∂disorderLaw h, ∃ H : Hilbert →L[ℂ] Hilbert, ⏎ IsAndersonOperator v H ∧ IsSelfAdjoint H ∧ spectrum ℂ H = spectralInterval h := by" ] }, { "clause": "In every fixed dimension d ≥ 3, sufficiently weak positive disorder gives purely absolutely continuous spectrum on a fixed open interval with nonzero spectral weight.", "status": "unlinked", "coveredBy": [], "manuscript": "Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026", "evidence": [ "preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/build/paper.tex:41-45: For each fixed dimension $d\\ge3$ and each fixed sufficiently small positive ⏎ disorder strength, we prove that the Anderson operator on $\\mathbb Z^d$ with ⏎ independent uniform site potentials almost surely has purely absolutely ⏎ continuous spectrum with nonzero weight on an open energy interval. The ⏎ interval may depend on $d$ but is independent of the disorder strength.", "lean/docs/261.md:5: - [Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder](../../preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf)", "lean/docs/261.md:9: The linked formalization proves a supporting spectral statement for the nearest-neighbor Anderson operator on $\\mathbb Z^2$. For every disorder strength $h>0$ with independent site potentials uniform on $[-h,h]$, it constructs the bounded self-adjoint operator almost surely and identifies its spectrum as the real interval $[-4-h,4+h]$." ] } ], "gap": "Both spectral-type claims: pure-point spectrum in d = 2 (declared) and absolutely continuous spectrum in d ≥ 3 (undeclared). The only formal content is the almost-sure spectrum as a set in d = 2.", "scopeNoteOnGap": "This selected statement identifies the spectral set. It does not assert the pure-point spectral type claimed in the accompanying paper. [quote covers d = 2 only; the scope note is silent on d ≥ 3]", "note": "No headline clause is stated, and the stated spectral-set identification is not implied by the headline (pure-point type does not fix the spectrum as a set) nor implies it, so DIFFERENT by the precedence — though a declared one: the note calls it \"a supporting spectral statement\" and says it \"does not assert the pure-point spectral type\"; it is not on the README support list, so not SUPPORT. Disagrees with the first reader's NARROWER — UNDECLARED (more severe: \"narrower\" needs the headline to imply the statement, and here it does not); their undeclared d ≥ 3 gap is UNLINKED under amendment 3.", "firstSuspicious": "The scope note calls the statement supporting, but lean/ComparatorChallenges/README.md does not list it among supporting-result comparisons, so under the registered rule it is not SUPPORT. A looser reader could class this family SUPPORT-ONLY; the d ≥ 3 omission is undeclared either way.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "PlanarAndersonSpectrum", "flag": "NARROWER: spectral set only — no pure-point statement (declared) and nothing for d ≥ 3 (undeclared)" } ], "flagReadings": null, "readings": [ "first/batch1.json", "second/second6.json" ] }, { "family": "262", "title": "Sharp finite-matrix Lieb–Thirring inequalities and all equality cases", "headline": "Proves the sharp one-dimensional Lieb–Thirring inequality for $`1/2\\lt \\gamma\\lt 3/2`$ and arbitrary finite-matrix potentials W ≥ 0 with $`\\int\\mathop{\\mathrm{tr}}\\nolimits (W^{\\gamma+1/2})\\lt \\infty`$: the optimal constant is the scalar one-bound-state value, independent of matrix size. All equality cases are direct sums, in one constant unitary basis, of scalar sech2 solitons with independent scales and centers, and zero channels.", "challenges": [ "LiebThirring" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves the sharp one-dimensional Lieb–Thirring inequality for $`1/2\\lt \\gamma\\lt 3/2`$", "status": "stated", "coveredBy": [ "LiebThirring" ], "evidence": [ "lean/ComparatorChallenges/LiebThirring.lean:82: ∀ γ : ℝ, 1 / 2 < γ → γ < 3 / 2 →", "lean/ComparatorChallenges/LiebThirring.lean:83: (∀ W : ℝ → ℝ, Admissible γ W →", "lean/ComparatorChallenges/LiebThirring.lean:84: negativeMoment γ W ≤ ENNReal.ofReal (sharpConstant γ) * potentialMass γ W) ∧" ], "note": "Scalar potentials (matrix size 1) only: W : ℝ → ℝ, a.e. ≥ 0, in L^{γ+1/2}; the full negative-eigenvalue moment as a supremum over finite orthonormal eigenfamilies in ℝ≥0∞." }, { "clause": "and arbitrary finite-matrix potentials W ≥ 0 with $`\\int\\mathop{\\mathrm{tr}}\\nolimits (W^{\\gamma+1/2})\\lt \\infty`$", "status": "unlinked", "coveredBy": [], "manuscript": "Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026", "evidence": [ "CONTENTS.md:6561: We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent $`1/2\\lt \\gamma\\lt 3/2`$.", "lean/ComparatorChallenges/LiebThirring.lean:57: (∀ᵐ x : ℝ, 0 ≤ W x) ∧ MemLp W (ENNReal.ofReal (γ + 1 / 2)) volume", "lean/docs/262.md:5: - [Sharp one-dimensional Lieb–Thirring constants](../../preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/paper.pdf)", "preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/build/sections/introduction.tex:53: The theorem concerns scalar potentials on the line;" ] }, { "clause": "the optimal constant is the scalar one-bound-state value", "status": "stated", "coveredBy": [ "LiebThirring" ], "evidence": [ "lean/ComparatorChallenges/LiebThirring.lean:85: optimalConstant γ = ENNReal.ofReal (sharpConstant γ) ∧", "lean/ComparatorChallenges/LiebThirring.lean:86: oneStateConstant γ = ENNReal.ofReal (sharpConstant γ) ∧" ], "note": "Scalar case; attainment at (r+1)sech²(rx) is also stated (lines 87-90)." }, { "clause": "independent of matrix size.", "status": "unlinked", "coveredBy": [], "manuscript": "Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026", "evidence": [ "CONTENTS.md:6561: The optimal constant is the scalar one-bound-state constant, independently of the matrix size.", "lean/docs/262.md:5: - [Sharp one-dimensional Lieb–Thirring constants](../../preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/paper.pdf)" ] }, { "clause": "All equality cases are direct sums, in one constant unitary basis, of scalar sech2 solitons with independent scales and centers, and zero channels.", "status": "unlinked", "coveredBy": [], "manuscript": "Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026", "evidence": [ "CONTENTS.md:6552: equality holds precisely for direct sums, in one constant unitary basis, of scalar one-bound-state solitons and zero channels.", "lean/docs/262.md:5: - [Sharp one-dimensional Lieb–Thirring constants](../../preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/paper.pdf)", "preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/build/sections/introduction.tex:54: Its sharpness assertion", "preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/build/sections/introduction.tex:55: exhibits an equality potential without classifying all optimizers." ], "note": "Not stated even for scalars (only attainment at one explicit potential), and the linked scalar paper itself says it does not classify optimizers, so the whole classification belongs to the unlinked October-5 manuscript." } ], "gap": "The matrix-valued inequality (arbitrary finite matrix size, constant independent of size) and the classification of all equality cases — the content of two of the family's three manuscripts and the subject of the family title — are not stated. Only the scalar sharp inequality with optimality and attainment at one sech² potential is.", "scopeNoteOnGap": "silent — docs/262.md lists only the scalar manuscript (line 5: 'Sharp one-dimensional Lieb–Thirring constants') and describes only scalar potentials (line 9: 'For every nonnegative $W\\in L^{\\gamma+1/2}(\\mathbb R)$, it bounds the full negative-eigenvalue moment…'); it never says the matrix case or the equality classification is left out.", "note": "The scalar sharp inequality, both optimal constants and attainment are stated; the finite-matrix inequality, size-independence and the classification of equality cases come from the two October-5 manuscripts docs/262.md does not link (it links only the September-23 scalar paper): NARROWER — UNLINKED. The first reader found the same gap and called it UNDECLARED before amendment 3; the difference is the amendment. Note the family TITLE (matrix inequalities, all equality cases) names exactly what is not stated.", "firstSuspicious": "The challenge opens Matrix.Norms.L2Operator and Matrix (line 18–19) but states nothing matrix-valued — consistent with a scalar statement standing in for a matrix headline.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json", "second/second7.json" ] }, { "family": "263", "title": "The ionization and generalized ionization conjectures", "headline": "For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with M fixed nuclei of charges at least one and total charge Z strictly binds at most $`Z+CM`$ electrons. Neutral-atom first ionization energies and radii containing all but an expected half-electron have universal positive upper and lower bounds. The energy cost of removing m electrons has Thomas–Fermi asymptotics as $`m\\to\\infty`$ and $`Z/m\\to\\infty`$; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking $`Z\\to\\infty`$ first.", "challenges": [ "CoulombIonization", "CoulombRadii" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with M fixed nuclei of charges at least one and total charge Z strictly binds at most $`Z+CM`$ electrons.", "status": "unlinked", "coveredBy": [], "manuscript": "Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026", "evidence": [ "corpus/openai-math/release.json:11106: \"dir\": \"Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026\"", "corpus/openai-math/release.json:11108: We prove that a molecule with M fixed nuclei of real charges at least one and total nuclear charge Z strictly binds at most $`Z+CM`$ electrons, where C is universal.", "lean/docs/263.md:3: The following describes the scope of the Lean formalization related to the following accompanying paper(s):", "lean/docs/263.md:5: - [Generalized ionization energies for full Coulomb atoms](../../preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf)", "lean/docs/263.md:6: - [Generalized outer-electron radii of neutral Coulomb atoms](../../preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026/paper.pdf)" ] }, { "clause": "Neutral-atom first ionization energies and radii containing all but an expected half-electron have universal positive upper and lower bounds.", "status": "unlinked", "coveredBy": [], "manuscript": "Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026", "evidence": [ "corpus/openai-math/release.json:11108: the first ionization energy and, for every ground state, the radius outside which one half of an electron remains are each bounded above and below by positive universal constants.", "lean/docs/263.md:5: - [Generalized ionization energies for full Coulomb atoms](../../preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf)", "lean/docs/263.md:6: - [Generalized outer-electron radii of neutral Coulomb atoms](../../preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026/paper.pdf)" ] }, { "clause": "The energy cost of removing m electrons has Thomas–Fermi asymptotics as $`m\\to\\infty`$ and $`Z/m\\to\\infty`$;", "status": "stated", "coveredBy": [ "CoulombIonization" ], "evidence": [ "lean/ComparatorChallenges/CoulombIonization.lean:72: def TFCharacterization (a : ℝ) : Prop :=", "lean/ComparatorChallenges/CoulombIonization.lean:73: ∀ m : ℝ, 0 < m →", "lean/ComparatorChallenges/CoulombIonization.lean:74: Tendsto (tfIonization m) atTop (𝓝 (a * m ^ (7 / 3 : ℝ)))", "lean/ComparatorChallenges/CoulombIonization.lean:75: ", "lean/ComparatorChallenges/CoulombIonization.lean:76: def JointLimit (a : ℝ) : Prop :=", "lean/ComparatorChallenges/CoulombIonization.lean:77: ∀ m Z : ℕ → ℕ, (∀ j, 1 ≤ m j ∧ m j < Z j) →", "lean/ComparatorChallenges/CoulombIonization.lean:78: Tendsto m atTop atTop →", "lean/ComparatorChallenges/CoulombIonization.lean:79: Tendsto (fun j => (Z j : ℝ) / (m j : ℝ)) atTop atTop →", "lean/ComparatorChallenges/CoulombIonization.lean:80: Tendsto (fun j => ionization (m j) (Z j) / (m j : ℝ) ^ (7 / 3 : ℝ))", "lean/ComparatorChallenges/CoulombIonization.lean:81: atTop (𝓝 a)", "lean/ComparatorChallenges/CoulombIonization.lean:89: def MainStatement : Prop :=", "lean/ComparatorChallenges/CoulombIonization.lean:90: ∃ a : ℝ, 0 < a ∧ TFCharacterization a ∧ JointLimit a ∧ IteratedLimits a" ] }, { "clause": "neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking $`Z\\to\\infty`$ first.", "status": "stated", "coveredBy": [ "CoulombRadii" ], "evidence": [ "lean/ComparatorChallenges/CoulombRadii.lean:77: theorem generalized_outer_radii", "lean/ComparatorChallenges/CoulombRadii.lean:78: (Ψ : ∀ N : ℕ, Wavefunction (N + 1))", "lean/ComparatorChallenges/CoulombRadii.lean:79: (hΨ : ∀ N : ℕ, IsNormalizedGroundState (N + 1) (Ψ N)) :", "lean/ComparatorChallenges/CoulombRadii.lean:80: Tendsto (fun m : ℕ => (((m : ℝ) ^ (1 / 3 : ℝ) : ℝ) : EReal) * upperRadius Ψ m)", "lean/ComparatorChallenges/CoulombRadii.lean:81: atTop (𝓝 (bTF : EReal)) ∧", "lean/ComparatorChallenges/CoulombRadii.lean:82: Tendsto (fun m : ℕ => (((m : ℝ) ^ (1 / 3 : ℝ) : ℝ) : EReal) * lowerRadius Ψ m)", "lean/ComparatorChallenges/CoulombRadii.lean:83: atTop (𝓝 (bTF : EReal)) := by", "encoding: ground states are hypotheses (hΨ, CoulombRadii.lean:79); their existence is Zhislin's theorem, not formalized, and the note says so", "lean/docs/263.md:13: Ground states are inputs to the statement; their existence is not asserted separately." ] } ], "gap": "Not stated: the uniform excess-charge bound Z + CM for molecules, and the universal two-sided bounds on neutral-atom first ionization energies and half-electron radii. These are the family's first manuscript, which contains the ionization conjecture of the family title.", "scopeNoteOnGap": "silent. docs/263.md links only the two generalized-ionization papers (lines 5-6) and describes only them (lines 10, 12-13). It never mentions the uniform-excess-charge paper, the Z + CM bound, or the first-ionization and half-electron radius bounds.", "note": "The two generalized-ionization clauses are stated (radii with ground states as named inputs, declared at 263.md:13). The excess-charge bound Z+CM and the universal first-ionization / half-electron-radius bounds — the ionization conjecture of the family title — come from the uniform-excess-charge manuscript, which the note does not link, hence UNLINKED. I differ from the first reader only through amendment 3.", "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "CoulombRadii", "flag": "VACUITY? (declared): hPsi requires a normalized minimizer for every neutral atom. If one failed to exist in this encoding, the theorem would be vacuous. Existence is Zhislin's theorem (true) but is not stated. docs/263.md:13: \"Ground states are inputs to the statement; their existence is not asserted separately.\" Lean: ComparatorChallenges/CoulombRadii.lean:51-54, 79." } ], "flagReadings": null, "readings": [ "first/batch7.json", "second/second1.json" ] }, { "family": "266", "title": "Exactly three mutually unbiased bases in dimension six", "headline": "Proves $`N(6)=3`$, resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ6, but four cannot. The exclusion is a complete certified computation under the stated binary64 arithmetic and compiler conditions. An independent companion proves the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture for order-six complex Hadamard matrices outside Tao's cubic equivalence class.", "challenges": [ "MUBSix", "HadamardCubeFiber" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves $`N(6)=3`$, resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ6, but four cannot.", "coveredBy": [ "MUBSix" ], "note": "Only N(6) ≤ 5 is stated. Exclusion of four (and three's attainability) is not stated; the scope note declares that the bound of three and the exclusion of four are not established. Existence of three MUBs (classical) is not mentioned." }, { "clause": "The exclusion is a complete certified computation under the stated binary64 arithmetic and compiler conditions.", "coveredBy": [], "note": "A claim about the manuscript's floating-point verification pipeline; not formalized (declared: 'its computer-assisted exclusion of four arbitrary bases')." }, { "clause": "An independent companion proves the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture for order-six complex Hadamard matrices outside Tao's cubic equivalence class.", "coveredBy": [ "MUBSix" ], "note": "First conjunct of fourier_and_family_bound." } ], "gap": "N(6) ≤ 3 (no four MUBs in ℂ⁶) and the attainability of three; Lean proves only ≤ 5.", "scopeNoteOnGap": "The selected statement does not establish the paper's upper bound of three or its computer-assisted exclusion of four arbitrary bases.", "note": null, "firstSuspicious": "Existence of three MUBs in ℂ⁶ (classical) is neither stated nor mentioned by the scope note; counted as covered by the declared gap's framing.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "MUBSix", "flag": "NARROWER (declared): family bound 5, not 3" } ], "flagReadings": null, "readings": [ "first/batch1.json" ] }, { "family": "267", "title": "Positive-temperature Bose–Einstein condensation and exact quantum depletion", "headline": "Proves Bose–Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature, independent of volume, with positive condensate fraction in the thermodynamic limit. At zero temperature, proves the Bogoliubov leading quantum-depletion law for hard spheres and fixed bounded nonnegative radial finite-range potentials of positive scattering length, taking the thermodynamic limit before the dilute limit.", "challenges": [ "HardSphere" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves Bose–Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature, independent of volume, with positive condensate fraction in the thermodynamic limit.", "status": "declared", "coveredBy": [], "manuscript": "Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026", "evidence": [ "lean/docs/267.md:11: It is uniform in the gas parameter within this range and concerns ground states, without a positive-temperature assertion.", "lean/docs/267.md:5: - [Ground-state condensation in the dilute hard-sphere gas](../../preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026/paper.pdf)", "CONTENTS.md:6704: there is a strictly positive temperature, independent of the volume, at which the exact canonical Gibbs state has a positive condensate fraction in the thermodynamic limit." ], "note": "Both DECLARED and UNLINKED fit: the note names the omission ('without a positive-temperature assertion'), and the clause comes from an October-5 manuscript the note does not link. I record the less severe, more informative DECLARED. No Gibbs state or temperature appears in HardSphere." }, { "clause": "At zero temperature, proves the Bogoliubov leading quantum-depletion law for hard spheres and fixed bounded nonnegative radial finite-range potentials of positive scattering length, taking the thermodynamic limit before the dilute limit.", "status": "unlinked", "coveredBy": [], "manuscript": "Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026 (hard spheres); Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026 (bounded potentials)", "evidence": [ "lean/docs/267.md:5: - [Ground-state condensation in the dilute hard-sphere gas]", "CONTENTS.md:6713: giving the depletion fraction $`\\frac{8}{3\\sqrt\\pi}\\sqrt{\\rho a^3}+o(\\sqrt{\\rho a^3})`$ for density ρ and hard-sphere exclusion distance a.", "CONTENTS.md:6722: We prove the Bogoliubov quantum-depletion asymptotic for the ground state of a three-dimensional Bose gas with a fixed bounded, nonnegative, radial interaction of finite range and positive scattering length a.", "preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026/build/sections/introduction.tex:65: We prove a positive fraction; we do not", "preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026/build/sections/introduction.tex:66: estimate the optimal depletion or show that the fraction tends to one", "lean/ComparatorChallenges/HardSphere.lean:380: c₀≤liminf (fun k ↦ occupation (hN k) (Ψ k).1) atTop) ∧" ], "note": "Stated is only a uniform ground-state lower bound liminf (condensate fraction) ≥ c₀ for ρa³ < ε₀, for hard spheres, pure and ground-supported mixed states. The hard-sphere depletion law implies this (for small ρa³ the depleted fraction is < 1/2), so the challenge is a strict weakening, not something DIFFERENT. Nothing is stated for soft potentials. The linked paper itself disclaims the depletion law, and both depletion manuscripts are unlinked." } ], "gap": "Neither headline claim is stated. What is stated — ground-state condensation with a uniform positive lower bound c₀ on the condensate fraction in the dilute hard-sphere gas (the 'Ground-state condensation' manuscript) — is weaker than the zero-temperature depletion law (sentence 2) and says nothing at positive temperature (sentence 1). The positive-temperature omission is declared; the depletion-law omission (hard spheres and bounded potentials) is not.", "scopeNoteOnGap": "partly declared: docs/267.md:11 'It is uniform in the gas parameter within this range and concerns ground states, without a positive-temperature assertion.' — silent on the Bogoliubov depletion law and on bounded soft potentials; docs/267.md:5 lists only the ground-state-condensation manuscript of the family's five.", "note": "Neither headline sentence is stated. The challenge is the linked September-24 ground-state condensation theorem, which the depletion law implies (so NARROWER, not DIFFERENT). The positive-temperature omission is declared in docs/267.md:11, and the depletion law comes from two unlinked October-5 manuscripts, so the family is UNLINKED. The first reader had the depletion gap as UNDECLARED, before amendment 3.", "firstSuspicious": "The family title is 'Positive-temperature Bose–Einstein condensation and exact quantum depletion', and the only challenge states neither: it is the September ground-state-condensation result. OpenAI's scope note discloses the missing temperature but not the missing depletion law.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json", "second/second2.json" ] }, { "family": "269", "title": "Uniform Laughlin gap and stability under bounded scalar disorder", "headline": "Proves the fermionic Laughlin spectral-gap conjecture for the full V1 interaction at filling 1/3 on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. Both the gap and disorder threshold are uniform over all sufficiently large particle numbers and all normalized potential profiles.", "challenges": [ "LaughlinGap", "Laughlin", "LaughlinFock", "LaughlinPlanar" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves the fermionic Laughlin spectral-gap conjecture for the full V1 interaction at filling 1/3 on the round sphere.", "coveredBy": [ "LaughlinGap", "Laughlin", "LaughlinFock", "LaughlinPlanar" ], "note": "∃ N₀ ∀ N ≥ N₀ ∀ antisymmetric ψ at flux Q = 3(N−1): γ·dist²(ψ, ℂ·Laughlin) ≤ ⟨ψ, H_{V1} ψ⟩ with γ = 1/25 (LaughlinGap) and 1/100 (Laughlin); Fock-space H² ≥ γH for γ < γ* = 0.0461… (LaughlinFock)." }, { "clause": "The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level.", "coveredBy": [], "note": "Not stated: no one-body potential appears." }, { "clause": "Both the gap and disorder threshold are uniform over all sufficiently large particle numbers and all normalized potential profiles.", "coveredBy": [ "LaughlinGap", "Laughlin" ], "note": "Only the unperturbed gap's uniformity in N (∃ N₀ ∀ N ≥ N₀); the disorder threshold and potential profiles are not stated." } ], "gap": "Stability of the gap and uniqueness of the ground state under projected bounded scalar one-body potentials (the October 5 'Uniform Stability' result).", "scopeNoteOnGap": "lean/docs/269.md:12: \"This selected statement is the unperturbed Fock-space inequality. Stability under projected one-body potentials and uniqueness of the perturbed ground state are outside it.\"", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "LaughlinGap", "flag": "ENCODING VERIFIED NUMERICALLY (by me, outside the clone): for Q ∈ {2,3,4,6,9,12} each pairCoefficient Q p (p < 2Q−1) is a unit vector in the two-particle orbital basis and an eigenvector of total J² with L = Q−1 (relative angular momentum 1) — so energy is exactly Σ_{i if h : i.val < n then x ⟨i.val, h⟩ else 0", "lean/ComparatorChallenges/QACParity.lean:59: ∑ y : Word N, if y out = parity x then Complex.normSq (W y (inputWord x)) else 0" ] }, { "clause": "Xu–Li's reductions give the same bounded-error obstruction for strict majority.", "status": "undeclared", "coveredBy": [], "manuscript": "Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026; Regular-trajectories-pruning-and-quantum-parity-September-24-2026", "classicalTheorem": "Xu–Li's fanout reductions (XuLi2026Fanout, Theorem 16 / Theorem 1 and Corollaries 2–3) — not formalized; no challenge mentions majority or symmetric functions.", "evidence": [ "preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/consequences.tex:274: \\begin{corollary}[Majority and symmetric functions]", "preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/consequences.tex:284: In particular, this excludes strict majority", "preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/consequences.tex:290: Xu and Li's Theorem~16 and Corollaries~2--3", "preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026/build/sections/07-parity.tex:158: exclude strict majority and broader symmetric families in our circuit", "lean/docs/274.md:5: - [Product-projection localization and the $\\mathrm{QAC}^0$ parity lower bound](../../preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf)", "lean/docs/274.md:6: - [Regular trajectories, pruning and quantum parity](../../preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026/paper.pdf)", "lean/docs/274.md:12: regular-trajectory propagation and pruning are not separate formalized statements here." ] } ], "gap": "The strict-majority lower bound (via Xu–Li's reductions); only parity is stated.", "scopeNoteOnGap": "silent", "note": "The parity lower bound in the measured-output model is stated exactly (any fixed advantage, polynomial total qubits, zero ancillas, garbage summed). The strict-majority consequence is a corollary in both linked manuscripts through Xu–Li's reductions, not stated, and docs/274.md names a different omission (regular-trajectory propagation and pruning) but never majority: UNDECLARED. Agrees with the first reader; the gap is a one-line corollary by citation, so it is the narrowest UNDECLARED kind.", "firstSuspicious": "The scope note (lines 1-21) never mentions majority. Mild: the headline itself attributes the majority bound to Xu–Li's reductions; under a rule that tolerates named external inputs this would read MATCHES.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch1.json", "second/second5.json" ] }, { "family": "275", "title": "QMA-hardness of continuum Coulomb energy", "headline": "Proves QMA-hardness of approximating the electronic Coulomb energy infimum in three dimensions, minimizing over the full spinful fermionic continuum space. Deterministic polynomial-time reductions work even with only unit-charge nuclei at distinct rational positions, polynomially many electrons and an energy-threshold separation of at least one.", "challenges": [ "ContinuumCoulombHardness" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves QMA-hardness of approximating the electronic Coulomb energy infimum in three dimensions, minimizing over the full spinful fermionic continuum space.", "coveredBy": [ "ContinuumCoulombHardness" ], "note": "ground energy = EReal sInf of the quadratic form over normalized antisymmetric H^1 states on (ℝ^3 × {spin})^n, all spin sectors" }, { "clause": "Deterministic polynomial-time reductions work even with only unit-charge nuclei at distinct rational positions, polynomially many electrons and an energy-threshold separation of at least one.", "coveredBy": [ "ContinuumCoulombHardness" ], "note": "unit_coulomb_qmaHard: unary electron count, distinct rational positions, thresholds with upper - lower >= 1; binary_coulomb_qmaHard gives the binary-charge companion" } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "ContinuumCoulombHardness", "flag": "JUNK? cleared: Coulomb terms are Bochner integrals (0 if non-integrable), but for H^1 states Hardy's inequality makes them integrable; coulombKernel 0 = 0⁻¹ = 0 only on a null set; ground energy uses EReal so sInf has no junk; BinaryRational.value divides by a denominator that Valid forces positive" }, { "challenge": "ContinuumCoulombHardness", "flag": "QMA definition uses TM2ComputableInPolyTime for the verifier generator without a FiniteAlphabet guard; since a TM2 machine with finitely many states can push only finitely many symbols this does not weaken polynomial time (contrast PerfectCompleteness, which adds the guard)" } ], "flagReadings": [ { "challenge": "ContinuumCoulombHardness", "flag": "JUNK? cleared: Coulomb terms are Bochner integrals (0 if non-integrable), but for H^1 states Hardy's inequality makes them integrable; coulombKernel 0 = 0⁻¹ = 0 only on a null set; ground energy uses EReal so sInf has no junk; BinaryRational.value divides by a denominator that Valid forces positive", "decision": "cleared", "why": "Checked each route. (1) The Coulomb terms are Bochner integrals: junk 0 would RAISE the form (dropping attraction) or lower it (dropping repulsion), a two-sided effect, so integrability matters. It holds: H1Vector is genuine H¹(ℝ^{3n}), with L² value, L² gradient and the weak-derivative identity against smooth compactly supported tests (lines 148-156). Hardy's inequality in each electron's coordinates (∫|ψ|²/|x_i−R|² ≤ 4∫|∇_iψ|², and likewise in x_i − x_j for pairs) plus Cauchy–Schwarz makes ∫ attraction·|ψ|² and ∫ pair·|ψ|² finite. (2) coulombKernel 0 = 0⁻¹ = 0 only on {x_i = R_j} ∪ {x_i = x_j}, a null set. (3) groundEnergy is an EReal sInf. ⊤ (empty set) would make every valid instance a NO instance, which blocks YES ↦ YES, so it is harmful to the claim and not helpful. And it does not occur: for every n ≥ 1, normalized antisymmetric H¹ states exist (Slater determinants with spin). (4) BinaryRational.value divides by denominators that Valid forces positive for every coordinate and for lower/upper. Instances that are not valid lie in neither promise set. Lane K: not run yet.", "evidence": [ "ComparatorChallenges/ContinuumCoulombHardness.lean:139: noncomputable def coulombKernel (x : Space) : ℝ := ‖x‖⁻¹", "ComparatorChallenges/ContinuumCoulombHardness.lean:148: structure H1Vector (n : ℕ) where", "ComparatorChallenges/ContinuumCoulombHardness.lean:149: value : Spins n → Configuration n → ℂ", "ComparatorChallenges/ContinuumCoulombHardness.lean:150: gradient : Spins n → (Fin n × Fin 3) → Configuration n → ℂ", "ComparatorChallenges/ContinuumCoulombHardness.lean:151: value_L2 : ∀ s, MemLp (value s) 2 volume", "ComparatorChallenges/ContinuumCoulombHardness.lean:152: partial_L2 : ∀ s a, MemLp (gradient s a) 2 volume", "ComparatorChallenges/ContinuumCoulombHardness.lean:153: weak_partial : ∀ s a (φ : Configuration n → ℝ),", "ComparatorChallenges/ContinuumCoulombHardness.lean:154: ContDiff ℝ (⊤ : ℕ∞) φ → HasCompactSupport φ →", "ComparatorChallenges/ContinuumCoulombHardness.lean:155: (∫ x, value s x * (fderiv ℝ φ x (EuclideanSpace.single a 1) : ℂ)) =", "ComparatorChallenges/ContinuumCoulombHardness.lean:156: -(∫ x, gradient s a x * (φ x : ℂ))", "ComparatorChallenges/ContinuumCoulombHardness.lean:169: noncomputable def nuclearEnergy {M n : ℕ} (S : Nuclei M) (ψ : H1Vector n) : ℝ :=", "ComparatorChallenges/ContinuumCoulombHardness.lean:170: ∑ s, ∫ x, (∑ i, attraction S (position x i)) * ‖ψ.value s x‖ ^ 2", "ComparatorChallenges/ContinuumCoulombHardness.lean:171: ", "ComparatorChallenges/ContinuumCoulombHardness.lean:172: noncomputable def pairEnergy {n : ℕ} (ψ : H1Vector n) : ℝ :=", "ComparatorChallenges/ContinuumCoulombHardness.lean:173: ∑ s, ∫ x, (∑ i : Fin n, ∑ j : Fin n,", "ComparatorChallenges/ContinuumCoulombHardness.lean:174: if i < j then coulombKernel (position x i - position x j) else 0) *", "ComparatorChallenges/ContinuumCoulombHardness.lean:175: ‖ψ.value s x‖ ^ 2", "ComparatorChallenges/ContinuumCoulombHardness.lean:176: ", "ComparatorChallenges/ContinuumCoulombHardness.lean:177: noncomputable def form {M n : ℕ} (S : Nuclei M) (ψ : H1Vector n) : ℝ :=", "ComparatorChallenges/ContinuumCoulombHardness.lean:178: kinetic ψ - nuclearEnergy S ψ + pairEnergy ψ", "ComparatorChallenges/ContinuumCoulombHardness.lean:426: def groundEnergy (d : NuclearData) : EReal :=", "ComparatorChallenges/ContinuumCoulombHardness.lean:427: sInf {e | ∃ state : FormState d.electrons,", "ComparatorChallenges/ContinuumCoulombHardness.lean:428: normSq state = 1 ∧ e = (coulombForm d state : EReal)}", "ComparatorChallenges/ContinuumCoulombHardness.lean:442: def BinaryPosition.Valid (p : BinaryPosition) : Prop :=", "ComparatorChallenges/ContinuumCoulombHardness.lean:443: 0 < p.x.denominator ∧ 0 < p.y.denominator ∧ 0 < p.z.denominator", "ComparatorChallenges/ContinuumCoulombHardness.lean:472: structure BinaryCoulomb.Valid (d : BinaryCoulomb) : Prop where", "ComparatorChallenges/ContinuumCoulombHardness.lean:473: nuclei_pos : 0 < d.nuclei.length", "ComparatorChallenges/ContinuumCoulombHardness.lean:474: electrons_pos : 0 < d.electrons", "ComparatorChallenges/ContinuumCoulombHardness.lean:475: positions : ∀ a : Fin d.nuclei.length, (d.nuclei.get a).1.Valid", "ComparatorChallenges/ContinuumCoulombHardness.lean:476: distinct : Function.Injective (fun a : Fin d.nuclei.length => (d.nuclei.get a).1.value)", "ComparatorChallenges/ContinuumCoulombHardness.lean:477: charge_pos : ∀ a : Fin d.nuclei.length, 0 < (d.nuclei.get a).2", "ComparatorChallenges/ContinuumCoulombHardness.lean:478: lower_denominator : 0 < d.lower.denominator", "ComparatorChallenges/ContinuumCoulombHardness.lean:479: upper_denominator : 0 < d.upper.denominator", "ComparatorChallenges/ContinuumCoulombHardness.lean:480: gap : 1 ≤ d.upper.value - d.lower.value", "ComparatorChallenges/ContinuumCoulombHardness.lean:526: def binaryCoulombPromise : PromiseProblem BinaryCoulomb where", "ComparatorChallenges/ContinuumCoulombHardness.lean:527: yes := {d | ∃ h : d.Valid, groundEnergy (d.toData h) ≤ ((d.lower.value : ℝ) : EReal)}", "ComparatorChallenges/ContinuumCoulombHardness.lean:528: no := {d | ∃ h : d.Valid, ((d.upper.value : ℝ) : EReal) ≤ groundEnergy (d.toData h)}" ] } ], "readings": [ "first/batch8.json", "flags/flags3.json" ] }, { "family": "276", "title": "Classical capacity of generalized amplitude damping", "headline": "Determines the unassisted classical capacity of every qubit generalized amplitude-damping channel, including all damping and thermal parameters. An explicit one-variable optimization gives the capacity, attained by independent two-state signal ensembles with collective decoding. Holevo capacity, minimum output entropy and regularized classical capacity are additive when tensoring with any finite-dimensional quantum channel.", "challenges": [ "AmplitudeDamping" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Determines the unassisted classical capacity of every qubit generalized amplitude-damping channel, including all damping and thermal parameters.", "coveredBy": [ "AmplitudeDamping" ], "note": "∀ γ, ν ∈ [0,1]: operational capacity = one-shot Holevo capacity." }, { "clause": "An explicit one-variable optimization gives the capacity, attained by independent two-state signal ensembles with collective decoding.", "coveredBy": [ "AmplitudeDamping" ], "note": "∃ maximiser p of h((1−γ)p + γν) − g(v(p)) on [0,1]; holevo n = n·objective(p)/log 2; every maximiser's 2^n equiprobable product phase-pair ensemble attains holevo n; capacity uses one collective POVM." }, { "clause": "Holevo capacity, minimum output entropy and regularized classical capacity are additive when tensoring with any finite-dimensional quantum channel.", "coveredBy": [], "note": "Not stated: only self-additivity holevo(n) = n·holevo(1) (declared)." } ], "gap": "Additivity with an arbitrary finite-dimensional partner channel (Holevo capacity, minimum output entropy, regularized capacity) is not stated.", "scopeNoteOnGap": "lean/docs/276.md:9: Additivity with an arbitrary different partner channel and decoding by separate measurements are not included.", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "277", "title": "Threshold repetition for entangled games", "headline": "Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction $`v+\\delta`$ of k independent repetitions decays exponentially in k, for $`0\\lt \\delta\\lt 1-v`$. Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed.", "challenges": [ "EntangledGames" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction $`v+\\delta`$ of k independent repetitions decays exponentially in k, for $`0\\lt \\delta\\lt 1-v`$", "coveredBy": [ "EntangledGames" ], "note": "There is a universal kappa0 > 0 such that for every game with v < 1, every 0 < delta < 1 - v and every k >= 1, P[wins >= ceil((v+delta)k)] <= exp(-kappa0 delta^13 k/(1 + log(|A||B|))), for every strategy and for the supremum over strategies." }, { "clause": "Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed", "coveredBy": [ "EntangledGames" ], "note": "Joint POVM strategies on (dimA+1) x (dimB+1) dimensional states for the k-fold game; questionProb is an arbitrary distribution on question pairs." } ], "gap": null, "scopeNoteOnGap": "no gap against the headline (see the rate flag)", "note": null, "firstSuspicious": "The Lean rate is delta^13 while the abstract claims delta^5. This does not touch the headline, but a page quoting the abstract's rate should say the formalized rate is weaker.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "EntangledGames", "flag": "RATE (abstract/paper, not headline): the abstract and the paper's Theorem 1.1 give rate delta^5 (preprints/.../build/main.tex:37; sections/00-introduction.tex:58: \"\\le \\exp\\!\\left(-\\frac{\\kappa_0\\delta^5}\"). The Lean states the weaker delta^13 (EntangledGames.lean:71). docs/277.md:9 states delta^13 without mentioning delta^5. The paper notes delta^13 follows (sections/00-introduction.tex:167-169, line 168: \"$\\kappa_0\\delta^{13}/(1+\\log d)$ already follows from\"). The headline says only \"decays exponentially in k\", so the family word is unaffected." } ], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "279", "title": "Exact quantum factoring over a fixed finite gate set", "headline": "Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices.", "challenges": [ "ExactQuantumFactoring" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one.", "coveredBy": [ "ExactQuantumFactoring" ], "note": "∃ family uniform (poly-time TM2 generator from 1^ℓ) with correctProbability = 1 for every N ≥ 2 of bit length ℓ." }, { "clause": "Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices.", "coveredBy": [ "ExactQuantumFactoring" ], "note": "PolynomialResources bounds qubits and instructions by one polynomial; 20 fixed gate names." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "280", "title": "Unitary vertex operator algebras and conformal nets", "headline": "Proves the strongly rational case of the strong-locality conjecture: every simple unitary strongly rational complex vertex operator algebra generates a completely rational conformal net. Its simple modules are unitarizable, and its representation category agrees with the net’s finite-index sectors as a braided unitary tensor category.", "challenges": [ "VertexAlgebraNet" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves the strongly rational case of the strong-locality conjecture:", "coveredBy": [ "VertexAlgebraNet" ], "note": "CKLWStrongLocal: polynomial energy bounds for every vector and A(I) ⊆ A(I')′ for every proper arc I." }, { "clause": "every simple unitary strongly rational complex vertex operator algebra generates a completely rational conformal net.", "coveredBy": [ "VertexAlgebraNet" ], "note": "The net generated by the smeared fields is an irreducible, diffeomorphism-covariant, positive-energy conformal net; complete rationality of the net is not stated (declared)." }, { "clause": "Its simple modules are unitarizable,", "coveredBy": [], "note": "Declared outside." }, { "clause": "and its representation category agrees with the net’s finite-index sectors as a braided unitary tensor category.", "coveredBy": [], "note": "Declared outside." } ], "gap": "Complete rationality of the net, unitarizability of simple modules, and the braided unitary tensor equivalence.", "scopeNoteOnGap": "The selected statement does not include complete rationality of the net, unitarizability of all simple modules, the braided tensor equivalence, or the classification of local extensions described in the paper.", "note": null, "firstSuspicious": "831-line statement read through its closure; the conformal-net structure and CKLW locality look faithful, but several defs carry junk fallbacks whose non-use rests on true analytic facts (closability of smeared fields in particular).", "confidence": "medium", "conditionalOnLaneK": null, "flags": [ { "challenge": "VertexAlgebraNet", "flag": "JUNK? (not triggered): smoothSmearedField is 0 unless PolynomialEnergyBounds and rapid Fourier decay hold — that would trivialize the net, but the theorem's first conjunct asserts PolynomialEnergyBounds and smooth periodic f has rapidly decaying coefficients" }, { "challenge": "VertexAlgebraNet", "flag": "JUNK? (not triggered, load-bearing): localGraphData keeps only smeared fields whose `.closure` is closed; Mathlib's LinearPMap.closure returns a non-closable operator unchanged, so non-closable fields would silently drop out, shrinking A(I) and easing locality. For a unitary VOA the invariant form gives densely defined adjoints, so the fields are closable" }, { "challenge": "VertexAlgebraNet", "flag": "JUNK? (not triggered): resolvent, flow and projectiveStandardAction are defined by h.choose with fallbacks 0, 0, 1; the defining existentials hold for self-adjoint generators and the Möbius action, and a fallback for projectiveStandardAction would force a one-dimensional Hilbert space (vacuum_unique), i.e. falsity rather than weakness" } ], "flagReadings": [ { "challenge": "VertexAlgebraNet", "flag": "JUNK? (not triggered): smoothSmearedField is 0 unless PolynomialEnergyBounds and rapid Fourier decay hold — that would trivialize the net, but the theorem's first conjunct asserts PolynomialEnergyBounds and smooth periodic f has rapidly decaying coefficients", "decision": "cleared", "why": "Both fallbacks to 0 are dead under the theorem's own first conjunct. smoothSmearedField is the real operator iff PolynomialEnergyBounds ∧ RapidCoefficients(f̂): conjunct 1 of `main` asserts A.PolynomialEnergyBounds, and the Fourier coefficients (fourierCoeffOn on [0,2π]) of a ContDiff ℝ ∞ 2π-periodic function decay faster than every power (integrate by parts k times), so RapidCoefficients holds for every SmoothCircleFunction. The inner fallback in smearedMap (else 0 unless ∀ v, Summable) is also dead: for v ∈ V, ‖f̂(m)·a_{m+d−1}v‖ ≤ ‖f̂(m)‖·C(1+|m+d−1|)^p·‖(1+L0)^k v‖ (the energy bound) is summable against rapid decay, and Completion V is complete, so the family is norm-summable. So if the theorem holds, every smeared field is the genuine tsum Σ f̂(m) a_{m+d−1}; if conjunct 1 failed, the theorem would be false. The net is never trivialized by this junk.", "evidence": [ "ComparatorChallenges/VertexAlgebraNet.lean:700-702: exact if h : ∀ v, Summable (fun m => family m v) then ⏎ SummableLinearFamily.tsumMap family h ⏎ else 0", "ComparatorChallenges/VertexAlgebraNet.lean:707-711: exact if h : A.PolynomialEnergyBounds ∧ OAI.StronglyRationalVOA.RapidCoefficients (OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleFunction.coefficient f) then ⏎ (OAI.MinimalVertex.Code.StronglyRationalVOA.HilbertCore.operator ⏎ (OAI.MinimalVertex.smearedMap A a d (And.left h a) (OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleFunction.coefficient f) ⏎ (And.right h))) ⏎ else 0", "ComparatorChallenges/VertexAlgebraNet.lean:155-157: noncomputable def PolynomialEnergyBound (a : V) : Prop := ⏎ ∃ C : ℝ, 0 < C ∧ ∃ p k : ℕ, ∀ (n : ℤ) (b : V), ⏎ ‖OAI.StronglyRationalVOA.mode A.fields a n b‖ ≤ C * (1 + |(n : ℝ)|) ^ p * ‖(A.energyShift ^ k) b‖", "ComparatorChallenges/VertexAlgebraNet.lean:201-202: noncomputable def RapidCoefficients (f : ℤ → ℂ) : Prop := ⏎ ∀ p : ℕ, Summable (fun n : ℤ => ‖f n‖*(1+|(n : ℝ)|)^p)", "ComparatorChallenges/VertexAlgebraNet.lean:223-226: structure SmoothCircleFunction where ⏎ toFun : ℝ → ℂ ⏎ periodic : Function.Periodic toFun (2*Real.pi) ⏎ smooth : ContDiff ℝ ∞ toFun", "ComparatorChallenges/VertexAlgebraNet.lean:824: A.PolynomialEnergyBounds ∧ OAI.MinimalVertex.CKLWStrongLocal U hSR.2.2 ∧" ] }, { "challenge": "VertexAlgebraNet", "flag": "JUNK? (not triggered, load-bearing): localGraphData keeps only smeared fields whose `.closure` is closed; Mathlib's LinearPMap.closure returns a non-closable operator unchanged, so non-closable fields would silently drop out, shrinking A(I) and easing locality. For a unitary VOA the invariant form gives densely defined adjoints, so the fields are closable", "decision": "cleared", "why": "The mechanism is real (Mathlib's LinearPMap.closure is the operator itself when not closable, so `∃ h : (closedSmoothField …).IsClosed` would drop a non-closable field and shrink A(I), easing the locality conjunct), but no field is non-closable under the hypotheses and conjunct 1. Each smeared field T has dense domain (HilbertCore.space V = image of V, dense in Completion V). The UnitaryStructure field `invariant` gives, for a ∈ grade d, ⟨a_n b, c⟩ = ⟨b, (−1)^d Σ_{j≤d} (1/j!) (L_1^j θa)_{2d−j−n−2} c⟩, a finite combination of modes. With energy bounds for every vector (conjunct 1) and rapid decay, S c := Σ_m conj(f̂(m))·(that combination at n = m+d−1) c converges for every c ∈ V and ⟨T b, c⟩ = ⟨b, S c⟩. So V ⊆ dom T*, T* is densely defined, and T is closable. Its .closure is then the genuine closed closure, and every homogeneous field with tsupport f ⊆ I enters localGraphData.", "evidence": [ "ComparatorChallenges/VertexAlgebraNet.lean:713-715: noncomputable def closedSmoothField (U : A.UnitaryStructure) (hc : A.toVertexAlgebra.C2Cofinite) ⏎ (a : V) (d : ℕ) (f : OAI.StronglyRationalVOA.SmoothCircleFunction) : Completion V →ₗ.[ℂ] Completion V := ⏎ (OAI.MinimalVertex.smoothSmearedField U hc a d f).closure", "ComparatorChallenges/VertexAlgebraNet.lean:719-721: {x | ∃ (a : V) (d : ℕ), a ∈ A.grade d ∧ ∃ (f : OAI.StronglyRationalVOA.SmoothCircleFunction), ⏎ tsupport (OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleFunction.onCircle f) ⊆ O ∧ ∃ (h : (OAI.MinimalVertex.closedSmoothField U hc a d f).IsClosed), ⏎ ∃ i j : Bool, x = (OAI.MinimalVertex.Code.StronglyRationalVOA.ClosedGraphData.entry (OAI.MinimalVertex.closedSmoothField U hc a d f) h i j)}", "ComparatorChallenges/VertexAlgebraNet.lean:169-174: invariant : ∀ d : ℕ, ∀ a ∈ A.grade d, ∀ n : ℤ, ∀ b c : V, ⏎ inner ℂ (OAI.StronglyRationalVOA.mode A.fields a n b) c = ⏎ inner ℂ b (((-1 : ℂ) ^ d) • ⏎ ∑ j ∈ Finset.range (d + 1), ((j.factorial : ℂ)⁻¹) • ⏎ OAI.StronglyRationalVOA.mode A.fields ((A.virasoroMode 1 ^ j) (pct a)) ⏎ (2 * (d : ℤ) - j - n - 2) c)", "ComparatorChallenges/VertexAlgebraNet.lean:183-185: noncomputable def _root_.OAI.MinimalVertex.Code.StronglyRationalVOA.HilbertCore.space (V : Type*) [NormedAddCommGroup V] [InnerProductSpace ℂ V] : ⏎ Submodule ℂ (Completion V) := ⏎ LinearMap.range (Completion.toComplₗᵢ : V →ₗᵢ[ℂ] Completion V).toLinearMap", "ComparatorChallenges/VertexAlgebraNet.lean:731-734: noncomputable def CKLWStrongLocal (U : A.UnitaryStructure) (hc : A.toVertexAlgebra.C2Cofinite) : Prop := ⏎ (∀ a : V, A.PolynomialEnergyBound a) ∧ ⏎ ∀ I : OAI.StronglyRationalVOA.ProperCircleArc, OAI.MinimalVertex.intervalAlgebra U hc I ≤ ⏎ (OAI.MinimalVertex.intervalAlgebra U hc (OAI.MinimalVertex.Code.StronglyRationalVOA.ProperCircleArc.complement I)).commutant" ] }, { "challenge": "VertexAlgebraNet", "flag": "JUNK? (not triggered): resolvent, flow and projectiveStandardAction are defined by h.choose with fallbacks 0, 0, 1; the defining existentials hold for self-adjoint generators and the Möbius action, and a fallback for projectiveStandardAction would force a one-dimensional Hilbert space (vacuum_unique), i.e. falsity rather than weakness", "decision": "cleared", "why": "None of the three existentials fails, so no fallback is reached. (1) resolvent: hamiltonian_selfAdjoint is Mathlib's IsSelfAdjoint for a LinearPMap (T† = T), which forces a dense domain (Mathlib's IsSelfAdjoint.dense_domain) and closedness. So T is a genuine self-adjoint operator, and for |Im z| ≥ 1 the bounded R = (T−z)⁻¹ satisfies (R y, y + z R y) ∈ graph T, uniquely since T−z is injective. (2) flow: boundedGenerator(r) = (r²/2)((T−ir)⁻¹ + (T+ir)⁻¹) = r²T/(T²+r²), and exp(it·B_n) → e^{itT} strongly by spectral dominated convergence, so the limit L exists, is unique, and equals e^{itT}. (3) projectiveStandardAction: every M ∈ SL(2,ℝ) acts on the circle by the doubled-angle Möbius map, an orientation-preserving smooth diffeomorphism with a smooth 2π-equivariant lift F satisfying Implements M F. Rel M x y fixes y mod 2π, so the class of F in Group (lifts modulo the kernel of the circle action) depends only on M, and −1 acts trivially. The lifts proof shows products implement products, so the homomorphism exists and is unique. The first reader's backstop also checks out: fallback 0 for flow contradicts rotation_flow (isometries on H ≠ 0, vacuum_norm = 1), and fallback 1 for projectiveStandardAction makes extends_mobius force mobius F to be scalar, hence trivial (PSL(2,ℝ) is perfect), hence by vacuum_unique dim H = 1. Both would be falsity, never weakness.", "evidence": [ "ComparatorChallenges/VertexAlgebraNet.lean:744-748: noncomputable def resolvent (T : H →ₗ.[ℂ] H) (_hs : IsSelfAdjoint T) (z : ℂ) (_hz : 1 ≤ |z.im|) : ⏎ H →L[ℂ] H := by ⏎ classical ⏎ exact if h : ∃ R : H →L[ℂ] H, ∀ y, (R y, y + z • R y) ∈ T.graph then h.choose ⏎ else 0", "ComparatorChallenges/VertexAlgebraNet.lean:750-754: noncomputable def boundedGenerator (T : H →ₗ.[ℂ] H) (hs : IsSelfAdjoint T) ⏎ (r : ℝ) (hr : 1 ≤ r) : H →L[ℂ] H := ⏎ ((r : ℂ)^2/2) • ⏎ (OAI.MinimalVertex.resolvent T hs ((r : ℂ)*Complex.I) (by simpa [abs_of_nonneg (by linarith : 0 ≤ r)] using hr) + ⏎ OAI.MinimalVertex.resolvent T hs (-((r : ℂ)*Complex.I)) (by simpa [abs_of_nonneg (by linarith : 0 ≤ r)] using hr))", "ComparatorChallenges/VertexAlgebraNet.lean:767-773: noncomputable def flow {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] ⏎ (T : H →ₗ.[ℂ] H) (hs : IsSelfAdjoint T) (t : ℝ) : H →L[ℂ] H := by ⏎ classical ⏎ exact if h : ∃ L : H →L[ℂ] H, ∀ x, ⏎ Tendsto (fun n => OAI.MinimalVertex.approximatingFlow T hs n t x) atTop (𝓝 (L x)) then ⏎ h.choose ⏎ else 0", "ComparatorChallenges/VertexAlgebraNet.lean:775-781: noncomputable def projectiveStandardAction : PSL(2,ℝ) →* (OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.Group) := by ⏎ classical ⏎ exact if h : ∃ L : PSL(2,ℝ) →* (OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.Group), ⏎ ∀ M : SL(2,ℝ), ∃ F : OAI.StronglyRationalVOA.SmoothCircleLift, ∃ hF : (OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.Implements M F), ⏎ L (QuotientGroup.mk M) = ⏎ QuotientGroup.mk (⟨F, M, hF⟩ : (OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.lifts)) then h.choose ⏎ else 1", "ComparatorChallenges/VertexAlgebraNet.lean:543-545: noncomputable def _root_.OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.Rel (M : SL(2,ℝ)) (x y : ℝ) : Prop := ⏎ (M 0 0*Real.cos (x/2)+M 0 1*Real.sin (x/2))*Real.sin (y/2)= ⏎ (M 1 0*Real.cos (x/2)+M 1 1*Real.sin (x/2))*Real.cos (y/2)", "ComparatorChallenges/VertexAlgebraNet.lean:592: carrier := {F | ∃ M : SL(2,ℝ), (OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.Implements M F)}", "ComparatorChallenges/VertexAlgebraNet.lean:607: noncomputable abbrev _root_.OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.Group := OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.lifts ⧸ ((OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleLift.circleAction).comp OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.lifts.subtype).ker", "ComparatorChallenges/VertexAlgebraNet.lean:794: extends_mobius : ∀ F, diffeomorphism ((OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.inclusion) (OAI.MinimalVertex.projectiveStandardAction F))=QuotientGroup.mk (mobius F)", "ComparatorChallenges/VertexAlgebraNet.lean:800-802: vacuum_norm : ‖vacuum‖=1 ⏎ vacuum_invariant : ∀ F, mobius F vacuum=vacuum ⏎ vacuum_unique : ∀ x : H, (∀ F, mobius F x=x) → ∃ c : ℂ, x=c • vacuum", "ComparatorChallenges/VertexAlgebraNet.lean:805: hamiltonian_selfAdjoint : IsSelfAdjoint hamiltonian", "ComparatorChallenges/VertexAlgebraNet.lean:807-809: rotation_flow : ∀ t : ℝ, ⏎ (mobius (QuotientGroup.mk (OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.rotationMatrix t))).toLinearIsometry.toContinuousLinearMap= ⏎ OAI.MinimalVertex.flow hamiltonian hamiltonian_selfAdjoint t" ] } ], "readings": [ "first/batch1.json", "flags/flags4.json" ] }, { "family": "281", "title": "QAOA attains the SK optimum in the thermodynamic-first limit", "headline": "Proves that QAOA approaches the ground-state energy of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity before circuit depth. For every accuracy, finite depth and deterministic angles independent of size and disorder achieve the required limiting expected energy per spin. This also yields leading-order optimal expected MaxCut values on large-degree random regular graphs, with size tending to infinity before degree.", "challenges": [ "SKValue", "SKFullSupport" ], "word": "DIFFERENT", "firstWord": "DIFFERENT", "secondWord": "DIFFERENT", "disagreement": null, "clauses": [ { "clause": "Proves that QAOA approaches the ground-state energy of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity before circuit depth.", "status": "declared", "coveredBy": [], "manuscript": "QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026", "evidence": [ "lean/docs/281.md:12: It also proves convergence of finite Gaussian coefficient sums to the curvature integral, which approaches the ground-state value as the terminal time tends to one. The selected statement covers these value and approximation results; it does not itself assert convergence of QAOA circuit energies.", "lean/ComparatorChallenges/SKValue.lean:163-170: theorem value_consequences : ⏎ (∀ (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ), ⏎ IsMinimizer W γ → IsDiffusion W γ X → ScalarValueConclusion W γ X) ∧ ⏎ (∀ (W : BrownianSpace) (γ : OrderParameter) (X : ℝ → W.Ω → ℝ) (T : ℝ), ⏎ 0 < T → T < 1 → IsMinimizer W γ → IsDiffusion W γ X → ⏎ FiniteGaussianConclusion W γ X T) ∧ ⏎ (∀ (W : BrownianSpace) (γ : OrderParameter), ⏎ IsMinimizer W γ → groundStateValue = parisi W γ) := by" ] }, { "clause": "For every accuracy, finite depth and deterministic angles independent of size and disorder achieve the required limiting expected energy per spin.", "status": "declared", "coveredBy": [], "manuscript": "QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026", "evidence": [ "lean/docs/281.md:12: It also proves convergence of finite Gaussian coefficient sums to the curvature integral, which approaches the ground-state value as the terminal time tends to one. The selected statement covers these value and approximation results; it does not itself assert convergence of QAOA circuit energies." ] }, { "clause": "This also yields leading-order optimal expected MaxCut values on large-degree random regular graphs, with size tending to infinity before degree.", "status": "undeclared", "coveredBy": [], "manuscript": "QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026", "evidence": [ "preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026/build/sections/01-introduction.tex:218-222: \\begin{corollary}[Regular MaxCut in successive limits] ⏎ \\label{cor:regular-maxcut} ⏎ For every $\\eta>0$ there are an integer $p\\ge1$, deterministic normalized ⏎ angles $\\gamma,\\beta\\in\\R^p$, and an integer $D_0\\ge2$ such that, for every ⏎ integer $D\\ge D_0$ and every finite simple unweighted $(D+1)$-regular graph", "preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026/build/sections/01-introduction.tex:239-243: \\begin{equation}\\label{eq:random-regular-maxcut} ⏎ \\lim_{D\\to\\infty} ⏎ \\limsup_{\\substack{n\\to\\infty\\\\ n(D+1)\\ \\mathrm{even}}} ⏎ \\Pr_G\\!\\left[\\sqrt D\\,\\Delta_D(G_{n,D+1})>\\eta\\right]=0. ⏎ \\end{equation}", "lean/docs/281.md:10: The linked formalization supplies variational identities for the Gaussian zero-field Sherrington–Kirkpatrick ground-state energy used in the paper's QAOA argument. Conditional on an admissible Parisi minimizer and its associated diffusion, it proves convergence of the finite-system ground-state energy, identifies its limit with the Parisi value, and gives equivalent terminal-martingale and integrated-curvature formulas.", "lean/docs/281.md:12: It also proves convergence of finite Gaussian coefficient sums to the curvature integral, which approaches the ground-state value as the terminal time tends to one. The selected statement covers these value and approximation results; it does not itself assert convergence of QAOA circuit energies.", "lean/docs/281.md:14: The formalization proves that every admissible integrable minimizer of the zero-temperature Parisi functional for the pure zero-field Sherrington–Kirkpatrick model has full relative Stieltjes support on $[0,1)$. Equivalently, the order parameter increases strictly between every two overlap values below one, so its support has no gap. The covariance normalization is $\\xi(t)=t^2/2$.", "lean/docs/281.md:16: The statement also constructs the associated diffusion and proves its selected self-consistency moment identities. It is conditional on the order parameter being a minimizer and does not separately assert existence of a minimizer." ] } ], "gap": "The whole headline. The challenges state (i) value identities for the zero-temperature SK model conditional on a Parisi minimizer and its diffusion — convergence of the finite-n ground-state energy, equality with the Parisi value, terminal-martingale and integrated-curvature formulas, convergence of finite Gaussian coefficient sums — and (ii) full support of every zero-temperature Parisi minimizer. These are ingredients of the QAOA argument; none mentions QAOA.", "scopeNoteOnGap": "lean/docs/281.md:12: \"The selected statement covers these value and approximation results; it does not itself assert convergence of QAOA circuit energies.\"; silent on the MaxCut clause", "note": "No clause is stated; the two statements are ingredients (and the full-support paper is not in the headline at all) that neither imply nor are implied by the QAOA headline, so DIFFERENT. The QAOA convergence omission is declared (281.md:12); the regular-MaxCut corollary of the linked paper (01-introduction.tex:218-243) is not mentioned by the note. Not SUPPORT-ONLY: neither challenge is on the README list (release.json marks SKFullSupport support:true only because tools/pin-openai-math.js:120 also matches /Support$/ on the name). Agree with the first reader.", "firstSuspicious": "DIFFERENT vs SUPPORT-ONLY: the scope note frames these as supporting ('supplies variational identities … used in the paper's QAOA argument'), but they are not in the README's supporting list. Reclassify together with 033 and 194 if needed.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "SKValue", "flag": "VACUITY? every conjunct is conditional on IsMinimizer W γ; existence of a minimizer in this OrderParameter class is not asserted (declared, 281.md:16) — if none existed the statement would be vacuous; existence is classical (zero-temperature Parisi formula) but outside the statement" }, { "challenge": "SKValue", "flag": "JUNK (harmless direction): gradient/curvature are deriv (0 where non-differentiable) and groundStateValue is limUnder; a junk value would falsify the asserted identities rather than make them trivial; phi's sSup is over a nonempty bounded set" }, { "challenge": "SKFullSupport", "flag": "VACUITY? conditional on hmin : IsMinimizer W γ (existence not asserted; declared)" } ], "flagReadings": null, "readings": [ "first/batch6.json", "second/second6.json" ] }, { "family": "287", "title": "Isomorphism of the free group factors", "headline": "Resolves the free group factor isomorphism problem: $`L(\\mathbb F_2)\\cong L(\\mathbb F_3)`$, and hence all interpolated free group factors, including $`L(\\mathbb F_\\infty)`$, are isomorphic. Their common factor has fundamental group $`\\mathbb R_{\\gt 0}`$.", "challenges": [ "InterpolatedFactors" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the free group factor isomorphism problem: $`L(\\mathbb F_2)\\cong L(\\mathbb F_3)`$, and hence all interpolated free group factors, including $`L(\\mathbb F_\\infty)`$, are isomorphic", "coveredBy": [ "InterpolatedFactors" ], "note": "For all r, s ∈ (1, ∞] there is a normal (ultraweakly bicontinuous), trace-preserving *-isomorphism between the interpolated factors: L(F_n) = double commutant of the left regular representation on ℓ²(F_n) for integer n, L(F_∞) for r = ⊤, and for non-integer r the corner of B(ℓ²ℕ) ⊗̄ L(F_2) cut by a projection of trace t = (r−1)^{-1/2} with the normalised trace (the standard amplification L(F_2)_t = L(F_{1+1/t²}))." }, { "clause": "Their common factor has fundamental group $`\\mathbb R_{\\gt 0}`$", "coveredBy": [], "note": "Not stated (no fundamental group is defined). It follows by a short argument from the stated isomorphisms, because the non-integer models are literally the amplifications L(F_2)_t, giving L(F_2)_t ≅ L(F_2) for all t outside a countable set, and a subgroup of ℝ_{>0} containing a co-countable set is all of ℝ_{>0}. Declared as a consequence not stated." } ], "gap": "The fundamental-group clause is not stated; it is a (short, unformalized) consequence of the stated isomorphisms.", "scopeNoteOnGap": "docs/287.md:9: …The paper's fundamental-group conclusion is a further consequence rather than a separate selected statement here.", "note": null, "firstSuspicious": "Like 252, the only omission is a classical/short consequence of what is stated; it is recorded as NARROWER — DECLARED for strictness. The theorem body is sorry in the challenge as usual; the depth is in whether the definitions are the standard ones — I found them standard.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "288", "title": "Kadison's similarity conjecture", "headline": "Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra to operators on a Hilbert space becomes a $`*`$-homomorphism after conjugation by a bounded invertible operator.", "challenges": [ "KadisonSimilarity", "UniformCommutator" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra to operators on a Hilbert space becomes a $`*`$-homomorphism after conjugation by a bounded invertible operator.", "coveredBy": [ "KadisonSimilarity" ], "note": "similarityTheorem, universe-polymorphic in the algebra and the Hilbert space." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "KadisonSimilarity", "flag": "STATEMENT-FILE METAPROGRAM: two `run_cmd Lean.modifyEnv … auxLemmasExt.setState env {}` lines (aux-lemma cache reset; no declarations added)" }, { "challenge": "KadisonSimilarity", "flag": "JUNK? (not triggered): offDiagonalSeminorm is a real sSup over projections in the commutant, bounded by ‖T‖ and containing 0" }, { "challenge": "UniformCommutator", "flag": "JUNK? (not triggered): commutatorSeminorm is a real sSup of a set bounded by 2‖Y‖ and containing 0" } ], "flagReadings": [ { "challenge": "KadisonSimilarity", "flag": "JUNK? (not triggered): offDiagonalSeminorm is a real sSup over projections in the commutant, bounded by ‖T‖ and containing 0", "decision": "cleared", "why": "The range is nonempty because e = 0 is a self-adjoint idempotent in the commutant. It is bounded above by ‖T‖: for a self-adjoint idempotent e, both e and 1 − e are orthogonal projections of norm ≤ 1, so ‖(1−e)Te‖ ≤ ‖T‖. Real.sSup of a nonempty set that is bounded above is the true supremum. Even the junk value 0 would only shrink the right side of UniversalHyperreflexivity to 0 (forcing T ∈ M), which makes the statement harder, not easier. Lane K has not run KadisonSimilarity; this decision does not depend on it.", "evidence": [ "lean/ComparatorChallenges/KadisonSimilarity.lean:85: {e : H →L[ℂ] H // e ∈ M.commutant ∧ star e = e ∧ e * e = e}", "lean/ComparatorChallenges/KadisonSimilarity.lean:88: sSup (Set.range fun e : CommutantProjection M =>", "lean/ComparatorChallenges/KadisonSimilarity.lean:89: ‖(1 - (e : H →L[ℂ] H)) * T * (e : H →L[ℂ] H)‖)", "lean/ComparatorChallenges/KadisonSimilarity.lean:94: Metric.infDist T (M : Set (K →L[ℂ] K)) ≤ 2 * C * offDiagonalSeminorm M T" ] }, { "challenge": "UniformCommutator", "flag": "JUNK? (not triggered): commutatorSeminorm is a real sSup of a set bounded by 2‖Y‖ and containing 0", "decision": "cleared", "why": "The set contains 0 (take a = 0 ∈ P) and is bounded above by 2‖Y‖, since ‖Ya − aY‖ ≤ 2‖Y‖‖a‖ ≤ 2‖Y‖, so sSup is the true supremum. A junk 0 would only make the right-hand side 0, which is harder. Lane K: CERTIFIED.", "evidence": [ "lean/ComparatorChallenges/UniformCommutator.lean:13: sSup {r : ℝ | ∃ a : K →L[ℂ] K, a ∈ P ∧ ‖a‖ ≤ 1 ∧ r = ‖Y * a - a * Y‖}", "lean/ComparatorChallenges/UniformCommutator.lean:33: C * commutatorSeminorm P Y * ‖matrixOperator h X‖ := by" ] } ], "readings": [ "first/batch1.json", "flags/flags1.json" ] }, { "family": "289", "title": "Strong Kadison–Kastler stability and its spatial boundaries", "headline": "Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit balls. Counterexamples show that near-identity conjugacy fails for one-sided near inclusions, and that arbitrarily close norm-separable C∗-algebras need not be ambiently unitarily conjugate.", "challenges": [ "StrongKadisonKastler" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit balls.", "status": "stated", "coveredBy": [ "StrongKadisonKastler" ], "evidence": [ "lean/ComparatorChallenges/StrongKadisonKastler.lean:20: Metric.hausdorffDist (unitBall M) (unitBall N)", "lean/ComparatorChallenges/StrongKadisonKastler.lean:32: ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧", "lean/ComparatorChallenges/StrongKadisonKastler.lean:33: ∀ (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H],", "lean/ComparatorChallenges/StrongKadisonKastler.lean:35: kkDistance M N < δ → NearConjugacy M N ε" ], "note": "Encoding: δ is chosen before H, uniformly over Hilbert spaces in one universe u (the theorem is universe-polymorphic); unit balls are nonempty and bounded, so hausdorffDist is the true distance, not a junk value; Mathlib von Neumann algebras contain the identity." }, { "clause": "Counterexamples show that near-identity conjugacy fails for one-sided near inclusions", "status": "unlinked", "coveredBy": [], "manuscript": "Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026", "evidence": [ "CONTENTS.md:7207: [Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings]", "CONTENTS.md:7209: Spatial embeddings of the source into the target exist, but every implementing unitary stays a fixed positive distance from the identity.", "lean/docs/289.md:5: - [Universal strong Kadison–Kastler stability](../../preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026/paper.pdf)" ] }, { "clause": "and that arbitrarily close norm-separable C∗-algebras need not be ambiently unitarily conjugate.", "status": "unlinked", "coveredBy": [], "manuscript": "Close-Separable-Cstar-Algebras-Without-Spatial-Conjugacy-October-5-2026", "evidence": [ "CONTENTS.md:7216: [Close Separable C*-Algebras Without Spatial Conjugacy]", "CONTENTS.md:7218: that are not conjugate by any unitary.", "lean/docs/289.md:5: - [Universal strong Kadison–Kastler stability](../../preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026/paper.pdf)" ] } ], "gap": "Both counterexample clauses (the family's two October 5 manuscripts) are not stated.", "scopeNoteOnGap": "silent. docs/289.md links only the stability paper (line 5) and describes only it (line 9). Nothing is said about the one-sided near-inclusion or separable C*-algebra counterexamples.", "note": "The stability theorem is stated as headlined; both counterexample clauses come from the two October-5 manuscripts docs/289.md does not link (it links only the stability paper): NARROWER — UNLINKED. The first reader found the same gap and called it UNDECLARED before amendment 3; the difference is the amendment.", "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "StrongKadisonKastler", "flag": "NARROWER — UNDECLARED: only the stability theorem; both counterexample families are absent. Lean: lines 31-35. Headline: \"Counterexamples show that ...\". Scope note: silent (docs/289.md:5, 9)." } ], "flagReadings": null, "readings": [ "first/batch7.json", "second/second7.json" ] }, { "family": "290", "title": "Relative bicentralizers and modular spectral recovery", "headline": "Proves Connes' bicentralizer conjecture for every type III1 factor with separable predual and every faithful normal state. More generally, for every inclusion $`N\\subset M`$ of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra $`P\\subset N`$ with $`P'\\cap c(M)=N'\\cap c(M)`$, resolving the relative bicentralizer conjecture.", "challenges": [ "BoundedRecovery" ], "word": "DIFFERENT", "firstWord": "DIFFERENT", "secondWord": "DIFFERENT", "disagreement": null, "clauses": [ { "clause": "Proves Connes' bicentralizer conjecture for every type III1 factor with separable predual and every faithful normal state.", "status": "declared", "coveredBy": [], "manuscript": "Bounded-recovery-for-modular-spectral-averages-September-23-2026", "evidence": [ "lean/docs/290.md:9: The absolute bicentralizer conjecture is not included.", "corpus/openai-math/release.json:11680: obtain an alternative proof of bicentralizer triviality for type III1 factors with separable predual." ] }, { "clause": "More generally, for every inclusion $`N\\subset M`$ of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra $`P\\subset N`$ with $`P'\\cap c(M)=N'\\cap c(M)`$, resolving the relative bicentralizer conjecture.", "status": "unlinked", "coveredBy": [], "manuscript": "Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026", "evidence": [ "corpus/openai-math/release.json:11667: \"dir\": \"Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026\"", "corpus/openai-math/release.json:11669: This resolves the relative bicentralizer conjecture in this setting.", "lean/docs/290.md:3: The following describes the scope of the Lean formalization related to the following accompanying paper(s):", "lean/docs/290.md:5: - [Bounded recovery for modular spectral averages](../../preprints/Bounded-recovery-for-modular-spectral-averages-September-23-2026/Bounded-recovery-for-modular-spectral-averages-September-23-2026.pdf)", "lean/docs/290.md:9: The absolute bicentralizer conjecture is not included." ] } ], "gap": "both headline clauses; the Lean states a bounded spectral-recovery lemma for modular spectral averages that neither implies nor is implied by either clause", "scopeNoteOnGap": "The absolute bicentralizer conjecture is not included. (docs/290.md:9) — declares clause 1 out; silent on clause 2 (the relative bicentralizer conjecture)", "note": "No clause is stated: BoundedRecovery is a spectral-recovery lemma that neither implies nor is implied by either bicentralizer statement, so DIFFERENT; the absolute conjecture is declared out, the relative one is from an unlinked manuscript. Not on the README support list. The first reader's VACUITY? is cleared here: H = ℂ, M = B(ℂ), xi = 1, calculus f ↦ f(0)·id, J = conj inhabits StandardModularData with ScalarCentralizer, and s = 0, T = id, h n = 1 meet every hypothesis. Agree with the first reader.", "firstSuspicious": "DIFFERENT by the pre-registered definition; not in the README's supporting-result list, although it is plainly a supporting lemma. The scope note declares the absolute conjecture out but is silent on the relative bicentralizer conjecture, which the headline also claims. Read a second time per the discipline.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "BoundedRecovery", "flag": "VACUITY? (low): StandardModularData is a bespoke structure; if the Tomita-graph identity at lines 60-62 were inconsistent with the halfGraph convention, no S would exist and the theorem would be vacuous. Read here, halfGraph is the graph of Δ^{1/2} (cutoffs e^{min(n,u)/2}) and the identity is S = JΔ^{1/2}, so standard forms inhabit it; but the scalar-centralizer hypothesis plus hpositive are not shown jointly satisfiable in the statement" } ], "flagReadings": null, "readings": [ "first/batch8.json", "second/second1.json" ] }, { "family": "291", "title": "Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability", "headline": "Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C∗-algebra, resolving this case of Szabó's conjecture without restrictions on trace dynamics. The family also proves the unital Toms–Winter conjecture, equating strict comparison, finite nuclear dimension and Jiang–Su stability in the simple separable unital infinite-dimensional nuclear setting.", "challenges": [ "UniformGamma" ], "word": "DIFFERENT", "firstWord": "DIFFERENT", "secondWord": "DIFFERENT", "disagreement": null, "clauses": [ { "clause": "Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C∗-algebra, resolving this case of Szabó's conjecture without restrictions on trace dynamics.", "status": "unlinked", "coveredBy": [], "manuscript": "Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026", "evidence": [ "lean/docs/291.md:5: - [Tracial projection methods and uniform property $\\Gamma$](../../preprints/Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026/paper.pdf)", "CONTENTS.md:7259: absorbs the trivial action on the Jiang–Su algebra up to cocycle conjugacy. No restriction is imposed on the action on the tracial-state simplex." ], "note": "No group action, cocycle conjugacy or Jiang–Su algebra appears in UniformGamma." }, { "clause": "The family also proves the unital Toms–Winter conjecture, equating strict comparison, finite nuclear dimension and Jiang–Su stability in the simple separable unital infinite-dimensional nuclear setting.", "status": "undeclared", "coveredBy": [], "manuscript": "Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026", "evidence": [ "CONTENTS.md:7286: Independently, strict comparison implies Jiang–Su absorption for simple, separable, unital, infinite-dimensional nuclear algebras, resolving the unital Toms–Winter conjecture.", "lean/docs/291.md:11: The linked comparison results also prove Jiang–Su absorption and uniform property $\\Gamma$ for simple separable unital infinite-dimensional nuclear algebras with strict comparison", "lean/OAI/Analysis/JiangSu/Main.lean:107: theorem unital_main" ], "note": "The linked manuscript claims the unital Toms–Winter resolution (strict comparison ⇒ Z-stability; the other legs are prior literature, and the nuclear-dimension leg is also in the unlinked Nuclear-dimension-and-Jiang-Su-stability-without-elementary-subquotients-September-23-2026). No challenge states it. Far from declaring the omission, docs/291.md:11 credits it to 'the linked comparison results', which appear to be a solution-tree theorem (lean/OAI/Analysis/JiangSu/Main.lean:107) outside every Comparator closure. Flag: CREDIT OUTSIDE THE CLOSURE." } ], "gap": "The only challenge (UniformGamma) states a different theorem from the 'Tracial projection methods' manuscript: real rank zero of the uniform tracial ultrapower of the uniform tracial completion implies uniform property Γ (Schafhauser–Tikuisis–White Problem XXI). Neither headline clause is stated, and the stated theorem is neither implied by nor implies them.", "scopeNoteOnGap": "Not declared. The note instead claims more: lean/docs/291.md:11: \"The linked comparison results also prove Jiang–Su absorption and uniform property $\\Gamma$ for simple separable unital infinite-dimensional nuclear algebras with strict comparison, ...\". No Comparator challenge states Jiang–Su absorption. A non-Comparator Lean file in the solution tree (lean/OAI/Analysis/JiangSu/Main.lean, theorem unital_main: strict comparison ⇒ Jiang–Su absorption) appears to be what the sentence refers to; it is outside every Comparator configuration, and covers neither nuclear dimension nor the equivariant clause.", "note": "No headline clause is stated. UniformGamma states a different theorem from the linked paper (tracial real rank zero of the uniform tracial ultrapower ⇒ uniform property Γ, via a single halving projection, with five Prop-valued construction classes existentially proved, not assumed). It neither implies nor is implied by the headline, hence DIFFERENT; the Toms–Winter clause is also an undeclared gap inside a linked manuscript. Flags: CREDIT OUTSIDE THE CLOSURE (docs/291.md:11, and the file's line-9 comment citing a solution-side main_iff_original for 'statement preservation'). Agrees with the first reader.", "firstSuspicious": "The scope note (line 11) asserts Jiang–Su absorption results that no Comparator challenge states. They seem to live in a non-Comparator file (OAI/Analysis/JiangSu/Main.lean), so Comparator never checks them. Worth a second reader and possibly a lane-K note.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "UniformGamma", "flag": "LARGE STATEMENT (608 lines): read through the closure of CurrentMain.main: MainClaim → the five Construction classes → Statement → TopologicallySimple, IsNuclear/CStarAlgebraNorm, StablyFinite, RealRankZero, FamilyUltrapower/familyNullIdeal/IsFamilyNull/familyTwoNorm/traceTwoNorm, UniformTracialCompletion/familyCauchyAlgebra/familyCauchyNullIdeal, completionTrace/cauchyTrace/pullbackTrace/ultraTrace/ultraTraceValue, familyLimitTrace/descendedTrace, familyUltrapowerMap, UniformPropertyGammaAt." }, { "challenge": "UniformGamma", "flag": "VACUITY? checked, cleared: the five Prop classes are existentially quantified in MainClaim (lines 592–598), so they must be proved, not assumed; each is a true C*-algebra fact (e.g. NormConstruction is the C*-identity for quotients by closed ideals)." }, { "challenge": "UniformGamma", "flag": "JUNK? checked, none: familyTwoNorm is an iSup in ℝ (0 if unbounded) but is bounded by ‖a‖, since tracial states are positive and unital." } ], "flagReadings": null, "readings": [ "first/batch5.json", "second/second2.json" ] }, { "family": "292", "title": "Kirchberg's $`\\mathcal O_2`$ norm-ultrapower embedding problem", "headline": "Constructs an explicit separable unital full group C∗-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear C∗-algebra, for any free ultrafilter on the natural numbers. Taking the target to be $`\\mathcal O_2`$ answers Kirchberg's norm-ultrapower embedding problem negatively.", "challenges": [ "NuclearUltrapower" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs an explicit separable unital full group C∗-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear C∗-algebra, for any free ultrafilter on the natural numbers.", "coveredBy": [ "NuclearUltrapower" ], "note": "∃ separable A with ι: G →* unitary A satisfying the full group C*-algebra universal property for the explicit G = ℤ[1/2]³ ⋊ (SL₃(ℤ) × ℤ), such that for every nonzero unital B that is nuclear (min = max tensor norms against every C) and every free ultrafilter ω there is no injective unital *-homomorphism A → B^ω." }, { "clause": "Taking the target to be $`\\mathcal O_2`$ answers Kirchberg's norm-ultrapower embedding problem negatively.", "coveredBy": [ "NuclearUltrapower" ], "note": "The statement also produces O (universal unital C*-algebra on two Cuntz isometries, i.e. O₂) and asserts no injective unital *-hom A → O^ω for every free ω." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "NuclearUltrapower", "flag": "VACUITY? (checked, not vacuous): the clause '∀ B, IsNuclear B → …' would be vacuous if no B satisfied IsNuclear; ℂ (and O₂) satisfy min = max, so the hypothesis is inhabited." }, { "challenge": "NuclearUltrapower", "flag": "JUNK? (checked, benign): minTensorNorm/maxTensorNorm are sSup over ℝ (lines 145-154); the sets are nonempty (zero representation) and bounded by Σ‖bᵢ‖‖cᵢ‖, so the sSups are genuine." } ], "flagReadings": [ { "challenge": "NuclearUltrapower", "flag": "VACUITY? (checked, not vacuous): the clause '∀ B, IsNuclear B → …' would be vacuous if no B satisfied IsNuclear; ℂ (and O₂) satisfy min = max, so the hypothesis is inhabited.", "decision": "cleared", "why": "IsNuclear.{0,0} B says min = max on B ⊗ C for every NonUnitalCStarAlgebra C in Type 0. Take B = ℂ, which is a UnitalCStar. Each x is 1 ⊗ c. Every spatial pair gives ‖ρ(1) ⊗ σ(c)‖ ≤ ‖c‖ (norm_mapL_le; ⋆-homs are contractive), with equality for ρ the identity of ℂ on ℂ and σ faithful (a Type-0 GNS space). Every commuting pair gives ‖ρ(1)σ(c)‖ ≤ ‖c‖, with equality for ρ(λ) = λ·1 and the same σ. So min = max = ‖c‖, and the hypothesis is inhabited (O₂ also qualifies). Because the Lean predicate only tests C in Type 0, it is at least as permissive as genuine nuclearity. Every nuclear B in Type 0 satisfies it, so the ∀ B clause is not narrower than the prose. The O₂ conjunct (lines 197-199) does not use IsNuclear at all.", "evidence": [ "lean/ComparatorChallenges/NuclearUltrapower.lean:158: def IsNuclear (B : Type u) [NonUnitalCStarAlgebra B] : Prop :=", "lean/ComparatorChallenges/NuclearUltrapower.lean:159: ∀ (C : Type v) [NonUnitalCStarAlgebra C] (x : B ⊗[ℂ] C),", "lean/ComparatorChallenges/NuclearUltrapower.lean:160: minTensorNorm.{u, v, max u v} x = maxTensorNorm.{u, v, max u v} x", "lean/ComparatorChallenges/NuclearUltrapower.lean:194: (∀ (B : UnitalCStar.{u}), IsNuclear.{u, u} B →", "lean/ComparatorChallenges/NuclearUltrapower.lean:195: ∀ (ω : Ultrafilter ℕ), IsFree ω →", "lean/ComparatorChallenges/NuclearUltrapower.lean:196: ¬ ∃ φ : A →⋆ₐ[ℂ] NormUltrapower B ω, Function.Injective φ) ∧", "lean/ComparatorChallenges/NuclearUltrapower.lean:197: ∃ (O : UnitalCStar.{u}) (s : Fin 2 → O), IsCuntzTwoAlgebra O s ∧" ] }, { "challenge": "NuclearUltrapower", "flag": "JUNK? (checked, benign): minTensorNorm/maxTensorNorm are sSup over ℝ (lines 145-154); the sets are nonempty (zero representation) and bounded by Σ‖bᵢ‖‖cᵢ‖, so the sSups are genuine.", "decision": "cleared", "why": "Both sets are nonempty, because the zero representations (0 is a non-unital ⋆-hom) give r = 0. Both are bounded above. For x = Σ bᵢ ⊗ cᵢ, ‖spatialEval H K ρ σ x‖ ≤ Σ ‖mapL (ρ bᵢ) (σ cᵢ)‖ ≤ Σ ‖ρ bᵢ‖‖σ cᵢ‖ ≤ Σ ‖bᵢ‖‖cᵢ‖, using Mathlib's TensorProduct.norm_mapL_le and the contractivity of ⋆-homomorphisms between C⋆-algebras. Likewise ‖Σ ρ(bᵢ)σ(cᵢ)‖ ≤ Σ ‖bᵢ‖‖cᵢ‖ for commuting pairs. Both sSup are therefore genuine. Because representations range over all Type-0 Hilbert spaces, including faithful and universal ones, the two suprema are the spatial and maximal C⋆-norms.", "evidence": [ "lean/ComparatorChallenges/NuclearUltrapower.lean:127: TensorProduct.lift (LinearMap.mk₂ ℂ (fun b c => TensorProduct.mapL (ρ b) (σ c))", "lean/ComparatorChallenges/NuclearUltrapower.lean:146: sSup {r | ∃ (H K : ComplexHilbert.{w})", "lean/ComparatorChallenges/NuclearUltrapower.lean:147: (ρ : B →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (σ : C →⋆ₙₐ[ℂ] (K →L[ℂ] K)),", "lean/ComparatorChallenges/NuclearUltrapower.lean:148: r = ‖spatialEval H K ρ σ x‖}", "lean/ComparatorChallenges/NuclearUltrapower.lean:152: sSup {r | ∃ (H : ComplexHilbert.{w})", "lean/ComparatorChallenges/NuclearUltrapower.lean:153: (ρ : B →⋆ₙₐ[ℂ] (H →L[ℂ] H)) (σ : C →⋆ₙₐ[ℂ] (H →L[ℂ] H)),", "lean/ComparatorChallenges/NuclearUltrapower.lean:154: (∀ b c, Commute (ρ b) (σ c)) ∧ r = ‖commutingEval H ρ σ x‖}" ] } ], "readings": [ "first/batch3.json", "flags/flags2.json" ] }, { "family": "293", "title": "Invariant projections, hyperinvariant subspaces, and transitive algebras", "headline": "Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type II1 factor.", "challenges": [ "FiniteFactor", "BackwardIntertwiners", "ContinuousCircleWeight", "HyperinvariantSubspaces", "IrrationalRotation", "ProductBrown" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator.", "coveredBy": [ "HyperinvariantSubspaces", "BackwardIntertwiners" ], "note": "For every separable infinite-dimensional complex Hilbert space H: ∃ T ≠ 0 with ‖T^n‖^{1/n} → 0 whose commutant is transitive (every closed subspace invariant under all A with AT = TA is ⊥ or ⊤), proper, and SOT-closed. The two files state the same FullClaim." }, { "clause": "The construction also gives operators with no nontrivial invariant projection in the hyperfinite type II1 factor.", "coveredBy": [ "IrrationalRotation", "ContinuousCircleWeight", "FiniteFactor" ], "note": "IrrationalRotation: for EVERY irrational θ ∈ (0,1), a continuous weight f (values in [0,1], single zero at 0, ∫ −log f = ∞) such that the weighted rotation U·π(f) lies in the irrational-rotation von Neumann algebra (the crossed product L^∞(T) ⋊_θ ℤ), is nonzero and quasinilpotent, and has no nontrivial invariant projection ((1−p)Tp = 0 ⇒ p ∈ {0,1}). ContinuousCircleWeight: the same for one θ. FiniteFactor: in some II₁ factor with separable predual (not stated hyperfinite)." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": "'Hyperfinite' is never stated. It is carried by the concrete crossed-product model (IrrationalRotation, ContinuousCircleWeight), whose identification with R is classical. FiniteFactor alone would only give 'some II₁ factor'. Clause 1 resolves the hyperinvariant subspace problem, a famous open problem; the statement reads correctly, but it deserves a second reader for that reason alone.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [ { "challenge": "IrrationalRotation", "flag": "LARGE STATEMENT (850 lines): read through the closure of both theorems: IsContinuousCircleWeight/HasNegInfiniteLogIntegral/negLogCost, Rotation.{Hilbert, U, V, representation, orbitWeight, algebra, weighted(Real), vacuum}, WeightedShift.{shift, diagonal, fieldDiagonal}, vonNeumannGenerated, HasOnlyTrivialInvariantProjections, NormQuasinilpotent, vectorFugledeKadisonDeterminant; and for product_realization: ProductModel.{Base, baseMeasure, rotation, logarithmicCost, weight, AdmissibleAnglesAndCosts, factor, multiplier, weightedOperator, traceVector, HasAllFiniteCoordinateSampling, triangularLaw…}, ProductOperatorProperties, IsTypeII1FactorWithVector, HasSeparablePredual, IsVectorBrownMeasure, ProductPolarStructure." }, { "challenge": "IrrationalRotation", "flag": "META: run_cmd Lean.modifyEnv … auxLemmasExt.setState env {} appears 11 times (lines 158, 255, 315, 381, 426, 473, 533, 637, 665, 752, 764). It is an aux-lemma cache reset with no visible effect on the statements, but a second reader should confirm." }, { "challenge": "IrrationalRotation", "flag": "JUNK? checked, none: logarithmicCost is an ℝ≥0∞ tsum (no summability junk); the weight is EReal.exp(−cost).toReal ∈ [0,1]." }, { "challenge": "ProductBrown", "flag": "LARGE STATEMENT (758 lines): read through the closure of ProductBrownClaim: SourceAngles, sourceStage(State/Choice/next), sourceBlockSize, sourceLogWeight, sourceOperator/sourceShift/sourceMultiplication/sourceDiagonal/sourceCocycle/sourceLog/coordinateFloor/coordinateRotate, sourceMeasure, sourceVacuum/sourceVacuumFunctional, FugledeKadisonDeterminant, logarithmCutoff, expLogPotential, IsBrownMeasure." }, { "challenge": "ProductBrown", "flag": "VACUITY? checked, none: SourceAngles is satisfiable (it is a nonempty interval per coordinate)." }, { "challenge": "ProductBrown", "flag": "JUNK? checked, cleared: sourceLog is a real tsum (0 if not summable); since Σθ_i < ∞, only finitely many floors are nonzero almost surely, so the sum is finite a.e. and the junk value is never relevant in L²." }, { "challenge": "ProductBrown", "flag": "META: one run_cmd aux-lemma reset (line 697)." } ], "flagReadings": [ { "challenge": "IrrationalRotation", "flag": "JUNK? checked, none: logarithmicCost is an ℝ≥0∞ tsum (no summability junk); the weight is EReal.exp(−cost).toReal ∈ [0,1].", "decision": "cleared", "why": "There is no summability junk. logarithmicCost is an ℝ≥0∞ tsum. Its terms are ofReal(c_i·⌊rep(x_i)+θ_i⌋) with ⌊·⌋ ∈ {0,1}, and the tsum is always defined, possibly ∞. weight = (EReal.exp(−cost)).toReal is 0 exactly when cost = ∞, which is the true value of e^(−∞), not junk, and weight ∈ [0,1] (proved in-file, lines 561-569). The claim additionally asserts 0 < weight a.e. and ∫⁻ negLogCost(weight) = ∞ (lines 825-831), so the model cannot degenerate. negLogCost maps weights in [0,1] to −log ≥ 0 with no ofReal truncation. In the circle theorem, IsContinuousCircleWeight has f ∈ [0,1] with a single zero (a null set), and its lintegral is genuine. The vector FK determinant uses CFC.log of T*T + ε·1, which is self-adjoint with spectrum ⊆ [ε,∞), so the cfc is not junk. Even if it were, log = 0 gives determinant 1 ≠ 0, falsifying `= 0` rather than trivializing it. Lane K: not run yet.", "evidence": [ "ComparatorChallenges/IrrationalRotation.lean:16: def negLogCost (t : ℝ) : ℝ≥0∞ :=", "ComparatorChallenges/IrrationalRotation.lean:17: if t = 0 then ∞ else ENNReal.ofReal (-Real.log |t|)", "ComparatorChallenges/IrrationalRotation.lean:18: ", "ComparatorChallenges/IrrationalRotation.lean:19: def HasNegInfiniteLogIntegral {X : Type*} [MeasurableSpace X]", "ComparatorChallenges/IrrationalRotation.lean:20: (μ : Measure X) (f : X → ℝ) : Prop :=", "ComparatorChallenges/IrrationalRotation.lean:21: (∫⁻ x, negLogCost (f x) ∂μ) = ∞", "ComparatorChallenges/IrrationalRotation.lean:30: def IsContinuousCircleWeight (f : Circle → ℝ) : Prop :=", "ComparatorChallenges/IrrationalRotation.lean:31: Continuous f ∧ (∀ x, 0 ≤ f x ∧ f x ≤ 1) ∧", "ComparatorChallenges/IrrationalRotation.lean:32: (∀ x, f x = 0 ↔ x = 0) ∧ HasNegInfiniteLogIntegral circleMeasure f", "ComparatorChallenges/IrrationalRotation.lean:428: def vectorFugledeKadisonDeterminant (ξ : H) (T : H →L[ℂ] H) : ℝ≥0∞ :=", "ComparatorChallenges/IrrationalRotation.lean:429: ⨅ ε : {r : ℝ // 0 < r}, ENNReal.ofReal", "ComparatorChallenges/IrrationalRotation.lean:430: (Real.exp ((vectorTrace ξ (CFC.log (star T * T + ε.val • (1 : H →L[ℂ] H)))).re / 2))", "ComparatorChallenges/IrrationalRotation.lean:501: def logarithmicCost (θ c : ℕ → ℝ) (x : Base) : ℝ≥0∞ :=", "ComparatorChallenges/IrrationalRotation.lean:502: ∑' i, ENNReal.ofReal (c i * (⌊circleRepresentative (x i) + θ i⌋ : ℤ))", "ComparatorChallenges/IrrationalRotation.lean:503: ", "ComparatorChallenges/IrrationalRotation.lean:504: def weight (θ c : ℕ → ℝ) (x : Base) : ℝ :=", "ComparatorChallenges/IrrationalRotation.lean:505: (EReal.exp (-(logarithmicCost θ c x : EReal))).toReal", "ComparatorChallenges/IrrationalRotation.lean:561: theorem weight_nonneg (θ c : ℕ → ℝ) (x : Base) : 0 ≤ weight θ c x :=", "ComparatorChallenges/IrrationalRotation.lean:562: ENNReal.toReal_nonneg", "ComparatorChallenges/IrrationalRotation.lean:564: theorem weight_le_one (θ c : ℕ → ℝ) (x : Base) : weight θ c x ≤ 1 := by", "ComparatorChallenges/IrrationalRotation.lean:822: def ProductModel.IsCounterexampleRealization (θ c : ℕ → ℝ) : Prop :=", "ComparatorChallenges/IrrationalRotation.lean:823: ProductModel.AdmissibleAnglesAndCosts θ c ∧", "ComparatorChallenges/IrrationalRotation.lean:824: ProductModel.HasAllFiniteCoordinateSampling θ ∧", "ComparatorChallenges/IrrationalRotation.lean:825: Filter.Eventually (fun x : ProductModel.Base => 0 < ProductModel.weight θ c x)", "ComparatorChallenges/IrrationalRotation.lean:826: (@MeasureTheory.ae ProductModel.Base", "ComparatorChallenges/IrrationalRotation.lean:827: (@MeasureTheory.Measure ProductModel.Base MeasurableSpace.pi)", "ComparatorChallenges/IrrationalRotation.lean:828: MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass", "ComparatorChallenges/IrrationalRotation.lean:829: ProductModel.baseMeasure) ∧", "ComparatorChallenges/IrrationalRotation.lean:830: @HasNegInfiniteLogIntegral ProductModel.Base MeasurableSpace.pi", "ComparatorChallenges/IrrationalRotation.lean:831: ProductModel.baseMeasure (ProductModel.weight θ c) ∧" ] }, { "challenge": "ProductBrown", "flag": "VACUITY? checked, none: SourceAngles is satisfiable (it is a nonempty interval per coordinate).", "decision": "cleared", "why": "ProductBrownClaim is ∀ θ, SourceAngles θ → …, so it would be vacuous only if SourceAngles were unsatisfiable. It is not: it asks 1/(m_s·K_s) ≤ θ_i ≤ 2/(m_s·K_s) coordinatewise, and m_s ≥ 100 > 0, K_s ≥ 1 > 0 (in-file theorems sourceBlockSize_pos, sourceCutoff_pos), so θ_i := 1/(m_s·K_s) satisfies it. The stage data come from Classical.choice over a type proved nonempty in-file (exists_sourceStageChoice). The explicit @-instances in SourceAngles are the standard real division and order (the in-file SourceAngles.nonneg proof uses them). Lane K: not run yet.", "evidence": [ "ComparatorChallenges/ProductBrown.lean:445: def SourceAngles (θ : SourceCoordinate → ℝ) : Prop :=", "ComparatorChallenges/ProductBrown.lean:446: let realGroup : GroupWithZero ℝ :=", "ComparatorChallenges/ProductBrown.lean:447: (@DivisionRing.toDivisionSemiring ℝ Real.instField.toDivisionRing).toGroupWithZero", "ComparatorChallenges/ProductBrown.lean:448: let realOrder : LE ℝ := @Preorder.toLE ℝ Real.partialOrder.toPreorder", "ComparatorChallenges/ProductBrown.lean:449: ∀ i : SourceCoordinate,", "ComparatorChallenges/ProductBrown.lean:450: let block : ℝ := @Nat.cast ℝ Real.instNatCast (sourceBlockSize i.1)", "ComparatorChallenges/ProductBrown.lean:451: let cutoff : ℝ := @Nat.cast ℝ Real.instNatCast (sourceStageState i.1).K", "ComparatorChallenges/ProductBrown.lean:452: let denominator : ℝ := @Mul.mul ℝ Real.instMul block cutoff", "ComparatorChallenges/ProductBrown.lean:453: let lower : ℝ :=", "ComparatorChallenges/ProductBrown.lean:454: @Div.div ℝ realGroup.toDiv (@One.one ℝ Real.instOne) denominator", "ComparatorChallenges/ProductBrown.lean:455: let upper : ℝ :=", "ComparatorChallenges/ProductBrown.lean:456: @Div.div ℝ realGroup.toDiv (@Nat.cast ℝ Real.instNatCast 2) denominator", "ComparatorChallenges/ProductBrown.lean:457: @LE.le ℝ realOrder lower (θ i) ∧ @LE.le ℝ realOrder (θ i) upper", "ComparatorChallenges/ProductBrown.lean:459: theorem sourceBlockSize_pos (s : ℕ) : (0 : ℝ) < sourceBlockSize s := by", "ComparatorChallenges/ProductBrown.lean:460: exact_mod_cast (show 0 < (sourceStage s).m by have := (sourceStage s).m_ge_100; omega)", "ComparatorChallenges/ProductBrown.lean:461: ", "ComparatorChallenges/ProductBrown.lean:462: theorem sourceCutoff_pos (s : ℕ) : (0 : ℝ) < (sourceStageState s).K := by", "ComparatorChallenges/ProductBrown.lean:463: exact_mod_cast (show 0 < (sourceStageState s).K by have := (sourceStageState s).K_ge; omega)", "ComparatorChallenges/ProductBrown.lean:738: def ProductBrownClaim : Prop :=", "ComparatorChallenges/ProductBrown.lean:739: ∀ (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ),", "ComparatorChallenges/ProductBrown.lean:740: IsBrownMeasure sourceVacuumFunctional", "ComparatorChallenges/ProductBrown.lean:741: (sourceOperator θ sourceLogWeight hθ.nonneg sourceLogWeight_nonneg) (Measure.dirac 0)" ] }, { "challenge": "ProductBrown", "flag": "JUNK? checked, cleared: sourceLog is a real tsum (0 if not summable); since Σθ_i < ∞, only finitely many floors are nonzero almost surely, so the sum is finite a.e. and the junk value is never relevant in L².", "decision": "cleared", "why": "sourceLog is a real tsum; where the series diverges it gives 0, so the cocycle is 1 instead of the true e^(−∞) = 0. This happens only on a null set. The diagonal is exp(−Σ' c_i·⌊y_i + θ_i⌋) with y_i = fract(x_i + kθ_i). Each θ_i ≤ 2/(m_s K_s) ≤ 1/50 < 1, so each floor is 0 or 1, and it is 1 with probability exactly θ_i, since y_i is uniform on [0,1) under the infinitePi of Lebesgue on [0,1). Σ_i θ_i ≤ Σ_s 2/K_s < ∞, because K_0 = 100 and K_{s+1} = sourceNextCutoff m_s > 2m_s ≥ 2^{s+2} (sourceNextCutoff_large, m_ge_pow). By Borel–Cantelli, for each k a.e. x has only finitely many nonzero terms, so the series is summable there. The union over countably many k is null for the product with counting measure. The multiplication operator depends only on the a.e. class of its symbol, so the junk is never seen by sourceOperator. FugledeKadisonDeterminant uses cfc of logarithmCutoff n (continuous) on CFC.abs, so it is not junk either. Lane K: not run yet.", "evidence": [ "ComparatorChallenges/ProductBrown.lean:153: def sourceNextCutoff (m : ℕ) : ℕ := Nat.ceil ((m : ℝ)^((3/4 : ℝ)*(m : ℝ)))", "ComparatorChallenges/ProductBrown.lean:222: m_ge_K : a.K ≤ m", "ComparatorChallenges/ProductBrown.lean:223: m_ge_pow : 2^(s+1) ≤ m", "ComparatorChallenges/ProductBrown.lean:224: m_ge_G : 4*a.G ≤ (m : ℝ)", "ComparatorChallenges/ProductBrown.lean:225: m_ge_100 : 100 ≤ m", "ComparatorChallenges/ProductBrown.lean:282: def sourceStageState : ℕ → SourceStageState", "ComparatorChallenges/ProductBrown.lean:283: | 0 => ⟨100, 1, 0, 0, 1, by norm_num, le_rfl, le_rfl, le_rfl⟩", "ComparatorChallenges/ProductBrown.lean:284: | s+1 => (selectedSourceStage s (sourceStageState s)).next", "ComparatorChallenges/ProductBrown.lean:299: def coordinateFloor {ι : Type*} (θ : ι → ℝ) (k : ℤ) (x : ι → ℝ) (i : ι) : ℤ :=", "ComparatorChallenges/ProductBrown.lean:300: Int.floor (x i + (k : ℝ)*θ i)", "ComparatorChallenges/ProductBrown.lean:312: def coordinateRotate {ι : Type*} (θ : ι → ℝ) (k : ℤ) (x : ι → ℝ) (i : ι) : ℝ :=", "ComparatorChallenges/ProductBrown.lean:313: Int.fract (x i + (k : ℝ)*θ i)", "ComparatorChallenges/ProductBrown.lean:314: ", "ComparatorChallenges/ProductBrown.lean:315: def sourceLog {ι : Type*} (θ c : ι → ℝ) (k : ℤ) (x : ι → ℝ) : ℝ :=", "ComparatorChallenges/ProductBrown.lean:316: ∑' i, c i * (coordinateFloor θ k x i : ℝ)", "ComparatorChallenges/ProductBrown.lean:317: ", "ComparatorChallenges/ProductBrown.lean:318: def sourceCocycle {ι : Type*} (θ c : ι → ℝ) (k : ℤ) (x : ι → ℝ) : ℝ :=", "ComparatorChallenges/ProductBrown.lean:319: Real.exp (-sourceLog θ c k x)", "ComparatorChallenges/ProductBrown.lean:359: abbrev sourceMeasure (ι : Type*) : Measure ((ι → ℝ) × ℤ) :=", "ComparatorChallenges/ProductBrown.lean:360: (Measure.infinitePi (fun _ : ι => unitIntervalMeasure)).prod Measure.count", "ComparatorChallenges/ProductBrown.lean:364: def sourceDiagonal {ι : Type*} (θ c : ι → ℝ) (z : (ι → ℝ) × ℤ) : ℂ :=", "ComparatorChallenges/ProductBrown.lean:365: sourceCocycle θ c 1 (coordinateRotate θ z.2 z.1)", "ComparatorChallenges/ProductBrown.lean:699: def FugledeKadisonDeterminant (τ : ProductOp → ℂ) (T : ProductOp) : ℝ≥0∞ :=", "ComparatorChallenges/ProductBrown.lean:700: let determinant := fun (Hilbert : Type) [NormedAddCommGroup Hilbert]", "ComparatorChallenges/ProductBrown.lean:701: [InnerProductSpace ℂ Hilbert] [CompleteSpace Hilbert] [Nontrivial Hilbert]", "ComparatorChallenges/ProductBrown.lean:702: (trace : (Hilbert →L[ℂ] Hilbert) → ℂ) (operator : Hilbert →L[ℂ] Hilbert) =>", "ComparatorChallenges/ProductBrown.lean:703: letI : Module ℝ (Hilbert →L[ℂ] Hilbert) := Module.complexToReal _", "ComparatorChallenges/ProductBrown.lean:704: letI : IsometricContinuousFunctionalCalculus ℝ (Hilbert →L[ℂ] Hilbert)", "ComparatorChallenges/ProductBrown.lean:705: IsSelfAdjoint := IsSelfAdjoint.instIsometricContinuousFunctionalCalculus", "ComparatorChallenges/ProductBrown.lean:706: ⨅ n : ℕ, ENNReal.ofReal", "ComparatorChallenges/ProductBrown.lean:707: (Real.exp ((trace (cfc (logarithmCutoff n) (CFC.abs operator))).re))" ] } ], "readings": [ "first/batch5.json", "flags/flags3.json" ] }, { "family": "294", "title": "Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness", "headline": "Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C∗-algebras can have a properly infinite minimal tensor product, with one factor $`C_r^*(\\mathbb F_2)`$.", "challenges": [ "KaplanskyQuasitrace", "KaplanskyStableFiniteness" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive.", "coveredBy": [ "KaplanskyQuasitrace" ], "note": "∃ separable unital C*-algebra with a normalized 2-quasitrace and positive contractions a,b with Re(τ(a+b)-τ(a)-τ(b)) >= 1/144 for EVERY normalized 2-quasitrace" }, { "clause": "As a consequence, two unital simple stably finite C∗-algebras can have a properly infinite minimal tensor product, with one factor $`C_r^*(\\mathbb F_2)`$.", "coveredBy": [ "KaplanskyStableFiniteness" ], "note": "SimpleStableTensorCounterexample: C*_r(F2) simple and stably finite, and a nontrivial simple stably finite M with M ⊗_min C*_r(F2) properly infinite" } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "KaplanskyStableFiniteness", "flag": "VACUITY? cleared: Nontrivial M (and B) is required, which excludes the zero algebra, where 1 = 0 would make 'properly infinite' trivially true" } ], "flagReadings": [ { "challenge": "KaplanskyStableFiniteness", "flag": "VACUITY? cleared: Nontrivial M (and B) is required, which excludes the zero algebra, where 1 = 0 would make 'properly infinite' trivially true", "decision": "cleared", "why": "Each existential excludes the zero algebra, where 1 = 0 would make ProperlyInfinite, StablyFinite and IsEmpty TracialState trivial. TraceFreeStablyFinite requires Nontrivial A. ReducedQuasitraceLoss and SimpleStableTensorCounterexample require Nontrivial B / Nontrivial M and an injective unital ρ : B →⋆ₐ (H →L[ℂ] H). Injectivity from a nontrivial algebra forces H ≠ 0, so ℓ²(F2, H) ≠ 0 and tensorAlgebra ρ, a StarSubalgebra containing the identity operator, is nontrivial. ProperlyInfinite (tensorAlgebra ρ) therefore asserts two genuine isometries with orthogonal ranges. '¬∃ normalized 2-quasitrace' is not vacuous either, since B itself is required to carry one. Algebra F2 contains the identity of ℓ²(F2) ≠ 0, so its CStarSimple and StablyFinite conjuncts are about a nontrivial algebra.", "evidence": [ "ComparatorChallenges/KaplanskyStableFiniteness.lean:122-123: def ProperlyInfinite (A : Type*) [CStarAlgebra A] : Prop := ⏎ ∃ s t : A, star s*s = 1 ∧ star t*t = 1 ∧ star s*t = 0", "ComparatorChallenges/KaplanskyStableFiniteness.lean:158-160: def StablyFinite (A : Type*) [CStarAlgebra A] : Prop := ⏎ ∀ n : ℕ, 0 < n → ∀ v : CStarMatrix (Fin n) (Fin n) A, ⏎ star v * v = 1 → v * star v = 1", "ComparatorChallenges/KaplanskyStableFiniteness.lean:167-171: def TraceFreeStablyFinite : Prop := ⏎ ∃ (A : Type) (inst : CStarAlgebra A), ⏎ letI := inst ⏎ Nontrivial A ∧ TopologicalSpace.SeparableSpace A ∧ StablyFinite A ∧ ⏎ IsEmpty (TracialState A)", "ComparatorChallenges/KaplanskyStableFiniteness.lean:200-212: def ReducedQuasitraceLoss : Prop := ⏎ ∃ (B : Type) (instB : CStarAlgebra B), ⏎ letI := instB ⏎ Nontrivial B ∧ TopologicalSpace.SeparableSpace B ∧ ⏎ (∃ τ : OneQuasitrace B, τ.Normalized ∧ τ.IsTwo) ∧ ⏎ ∃ (H : Type) (instH : NormedAddCommGroup H) (innerH : InnerProductSpace ℂ H), ⏎ letI := instH ⏎ letI := innerH ⏎ ∃ (completeH : CompleteSpace H), ⏎ letI := completeH ⏎ ∃ ρ : B →⋆ₐ[ℂ] (H →L[ℂ] H), Function.Injective ρ ∧ ⏎ ProperlyInfinite (tensorAlgebra (G:=F2) ρ) ∧ ⏎ ¬ ∃ τ : OneQuasitrace (tensorAlgebra (G:=F2) ρ), τ.Normalized ∧ τ.IsTwo", "ComparatorChallenges/KaplanskyStableFiniteness.lean:222-233: def SimpleStableTensorCounterexample : Prop := ⏎ CStarSimple (Algebra F2) ∧ StablyFinite (Algebra F2) ∧ ⏎ ∃ (M : Type) (instM : CStarAlgebra M), ⏎ letI := instM ⏎ Nontrivial M ∧ CStarSimple M ∧ StablyFinite M ∧ ⏎ ∃ (H : Type) (instH : NormedAddCommGroup H) (innerH : InnerProductSpace ℂ H), ⏎ letI := instH ⏎ letI := innerH ⏎ ∃ (completeH : CompleteSpace H), ⏎ letI := completeH ⏎ ∃ ρ : M →⋆ₐ[ℂ] (H →L[ℂ] H), Function.Injective ρ ∧ ⏎ ProperlyInfinite (tensorAlgebra (G:=F2) ρ)" ] } ], "readings": [ "first/batch8.json", "flags/flags4.json" ] }, { "family": "295", "title": "The Kadison–Ringrose cohomology conjecture", "headline": "Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjecture.", "challenges": [ "KadisonRingrose" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive.", "coveredBy": [ "KadisonRingrose" ], "note": "Stated directly for degree n+2, n ≥ 0." }, { "clause": "Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjecture.", "coveredBy": [ "KadisonRingrose" ], "note": "The vanishing in degrees ≥ 2 is stated; the conjecture's degree-one case (derivations inner, Kadison–Sakai) is classical and declared outside the selected statement." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch1.json" ] }, { "family": "296", "title": "The generator problem for finite factors", "headline": "Proves that every type II1 factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion $`P\\subset M`$ of such factors, the unitaries u with $`M=W^*(P,u)`$ form a dense Gδ subset in the trace 2-norm topology.", "challenges": [ "FactorGeneration", "RelativeGeneration" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that every type II1 factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem.", "coveredBy": [ "FactorGeneration" ], "note": "Every II₁ factor S ⊆ B(H) with separable predual has x ∈ S with W*(x) = S. 'Equivalently by two self-adjoint operators' is the trivial x = Re x + i Im x equivalence and is not separately stated." }, { "clause": "More strongly, for every irreducible inclusion $`P\\subset M`$ of such factors, the unitaries u with $`M=W^*(P,u)`$ form a dense Gδ subset in the trace 2-norm topology.", "coveredBy": [ "RelativeGeneration" ], "note": "For II₁ factors small ≤ large with trivial relative commutant and large having separable predual: ∃ faithful normal normalized trace τ such that {u unitary in large | W*(small ∪ {u}) = large} is a dense G_δ in the topology generated by ‖·‖_{2,τ}-balls." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch3.json" ] }, { "family": "297", "title": "A ZFC counterexample to Naimark's problem", "headline": "Gives an alternative to [Tanaka's ZFC construction](https://arxiv.org/abs/2609.26930v1) of a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state and exactly one nonzero irreducible representation up to unitary equivalence. Thus the unrestricted compact-operator characterization fails without additional set-theoretic assumptions; the counterexample is nonseparable.", "challenges": [ "Naimark" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Gives an alternative to [Tanaka's ZFC construction](https://arxiv.org/abs/2609.26930v1) of a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state and exactly one nonzero irreducible representation up to unitary equivalence.", "coveredBy": [ "Naimark" ], "note": "∃ A (a CStarAlgebra, hence unital in Mathlib) with IsSimpleCStar, ¬FiniteDimensional, a faithful tracial state, and HasUniqueIrreducibleRepresentation (any two irreducible representations are unitarily equivalent). 'Exactly one' = at most one (stated) + at least one (every nonzero C*-algebra has one; not stated but classical). This holds only if the seven definitions keep their displayed bodies (DEFINITION HOLE)." }, { "clause": "Thus the unrestricted compact-operator characterization fails without additional set-theoretic assumptions; the counterexample is nonseparable.", "coveredBy": [ "Naimark" ], "note": "∀ H: ¬IsIsomorphicToCompacts A H; permitted axioms are propext, Quot.sound, Classical.choice, with no diamond-type axiom. Nonseparability is not stated, but it is forced: Rosenberg proved Naimark's problem positively for separable C*-algebras." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": "Every predicate in the statement is a definition hole. Under Comparator alone the theorem can be made trivial. MATCHES holds only for the displayed bodies and is conditional on lane K.", "confidence": "medium", "conditionalOnLaneK": [ { "challenge": "Naimark", "strict": "SAME BODIES" } ], "flags": [ { "challenge": "Naimark", "flag": "DEFINITION HOLE ×7: OAI.NaimarkZFC.IsSimpleCStar, IsState, IsFaithfulTracialState, IsIrreducible, UnitarilyEquivalent, HasUniqueIrreducibleRepresentation and IsIsomorphicToCompacts are all in definition_names. Comparator checks only their types. A solution could set, e.g., HasUniqueIrreducibleRepresentation := True and IsIsomorphicToCompacts := False, and the theorem would reduce to the existence of an infinite-dimensional C*-algebra, which is trivial. Every word of the conclusion is carried by holes, so lane K's strict run must decide the bodies before this row stands." } ], "flagReadings": null, "readings": [ "first/batch5.json" ] }, { "family": "298", "title": "Two notions of free entropy differ even when both are finite", "headline": "Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy $`-\\infty\\lt \\chi\\lt \\chi^*\\lt \\infty`$. This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.", "challenges": [ "FiniteEntropySeparation" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy $`-\\infty\\lt \\chi\\lt \\chi^*\\lt \\infty`$.", "coveredBy": [ "FiniteEntropySeparation" ], "note": "⊥ < χ(X) <= χ*(X) - 1/2 < ⊤, with n >= 2 variables" }, { "clause": "This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.", "coveredBy": [ "FiniteEntropySeparation" ], "note": "restatement" } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "FiniteEntropySeparation", "flag": "JUNK? cleared: every potentially singular quantity is EReal/ENNReal-valued (log 0 = ⊥, empty ⨅ = ⊤, lintegral never junk-0 by non-integrability), and the statement's own ⊥ < χ and χ* - 1/2 < ⊤ exclude the infinite values" }, { "challenge": "FiniteEntropySeparation", "flag": "JUNK? (low): ∫⁻ of a non-measurable integrand is a lower integral; s ↦ Φ*(X + sqrt(s) S) is monotone in s, so it is measurable" } ], "flagReadings": [ { "challenge": "FiniteEntropySeparation", "flag": "JUNK? cleared: every potentially singular quantity is EReal/ENNReal-valued (log 0 = ⊥, empty ⨅ = ⊤, lintegral never junk-0 by non-integrability), and the statement's own ⊥ < χ and χ* - 1/2 < ⊤ exclude the infinite values", "decision": "cleared", "why": "χ: ENNReal.log of the microstate volume is ⊥ only when the volume is genuinely 0 (the set lies in an operator-norm ball, so the volume is finite), which is Voiculescu's log 0 = −∞ convention. ⨅, ⨆ and limsup are taken in the complete lattice EReal. freeFisher: the empty ⨅ is ⊤, which is Φ* = ∞ when no conjugate system exists. χ*: the integrand is split into the ENNReal positive and negative parts (b − f) and (f − b), with truncated subtraction. These are exactly the positive and negative parts of n/(1+s) − Φ*(X+√s S), including f = ∞, where (b−f) = 0 and (f−b) = ∞. lintegral has no non-integrability junk, and toEReal sends ∞ to ⊤, not 0. The statement's own bounds exclude infinite values. ⊥ < χ ≤ χ* − 1/2 forces χ* > ⊥, hence ∫(f−b) < ∞, because otherwise χ* = ⊥ (in EReal, x − ⊤ = ⊥ and ⊤ − ⊤ = ⊥). χ* − 1/2 < ⊤ then forces ∫(b−f) < ∞. With both finite, χ* is the real number (n/2)log(2πe) + ½∫(n/(1+s) − Φ*) ds. Lane K has not run FiniteEntropySeparation.", "evidence": [ "lean/ComparatorChallenges/FiniteEntropySeparation.lean:62: ⨅ (ξ : Fin n → H) (_ : ConjugateSystem τ X ξ), ENNReal.ofReal (∑ i, ‖ξ i‖ ^ 2)", "lean/ComparatorChallenges/FiniteEntropySeparation.lean:72: (1 / 2 : EReal) * (∫⁻ s in Set.Ioi (0 : ℝ), b s - f s).toEReal -", "lean/ComparatorChallenges/FiniteEntropySeparation.lean:73: (1 / 2 : EReal) * (∫⁻ s in Set.Ioi (0 : ℝ), f s - b s).toEReal", "lean/ComparatorChallenges/FiniteEntropySeparation.lean:116: (⊥ : EReal) < chi τ X ∧", "lean/ComparatorChallenges/FiniteEntropySeparation.lean:117: chi τ X ≤ chiStar τ X S - (1 / 2 : EReal) ∧", "lean/ComparatorChallenges/FiniteEntropySeparation.lean:118: chiStar τ X S - (1 / 2 : EReal) < (⊤ : EReal) := by" ] }, { "challenge": "FiniteEntropySeparation", "flag": "JUNK? (low): ∫⁻ of a non-measurable integrand is a lower integral; s ↦ Φ*(X + sqrt(s) S) is monotone in s, so it is measurable", "decision": "cleared", "why": "The first reader's argument holds. f(s) = Φ*(X + √s S) is antitone in s, for three reasons. First, Φ* here depends only on the joint *-distribution: the conjugate-system identities and polynomialL2 are fixed by the moments under a faithful tracial vector. Second, for t > s, X + √t S has the distribution of (X + √s S) + √(t−s) S′, with S′ a free semicircular family free from (X, S): FreeStandardNoise makes the S i semicircular and free from each other and from X. Third, Voiculescu's inequality Φ*(Y + Z) ≤ Φ*(Y) for Y, Z free gives f(t) ≤ f(s). For s ≤ 0, √s = 0 and f(s) = Φ*(X) ≥ f(t). An antitone map ℝ → ℝ≥0∞ is measurable, and b is continuous, so b − f and f − b are measurable and both lintegrals are true integrals, not lower integrals. This rests on the standard free-probability inequality, not on anything run.", "evidence": [ "lean/ComparatorChallenges/FiniteEntropySeparation.lean:70: let f : ℝ → ℝ≥0∞ := fun s => freeFisher τ (heatTuple X S s)", "lean/ComparatorChallenges/FiniteEntropySeparation.lean:72: (1 / 2 : EReal) * (∫⁻ s in Set.Ioi (0 : ℝ), b s - f s).toEReal -" ] } ], "readings": [ "first/batch8.json", "flags/flags1.json" ] }, { "family": "299", "title": "The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability", "headline": "A nonzero unital separable complex C∗-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter. This answers the Kirchberg–Rørdam character question. Also, the infinite minimal tensor power of every such algebra without characters is Jiang–Su stable, answering the Dadarlat–Toms question.", "challenges": [ "CharacterCriterion" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "A nonzero unital separable complex C∗-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter.", "status": "stated", "coveredBy": [ "CharacterCriterion" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/CharacterCriterion.lean:2129: (A : Type) [CStarAlgebra A] [Nontrivial A] [TopologicalSpace.SeparableSpace A]", "lean/ComparatorChallenges/CharacterCriterion.lean:2130: (ω : Ultrafilter ℕ) (hω : (ω : Filter ℕ)≤Filter.cofinite) :", "lean/ComparatorChallenges/CharacterCriterion.lean:2131: OAI.KirchbergRordam.HasNoCharacters (NormUltrapower.CentralAlgebra A ω) ↔", "lean/ComparatorChallenges/CharacterCriterion.lean:2132: Nonempty (A ≃⋆ₐ[ℂ] MinTensor.Algebra A JiangSu.Algebra) := by sorry", "lean/ComparatorChallenges/CharacterCriterion.lean:2116: def Algebra : Type := CStarInductiveLimit.Algebra StandardPrimeModel.presentation.system" ] }, { "clause": "This answers the Kirchberg–Rørdam character question.", "status": "stated", "coveredBy": [ "CharacterCriterion" ], "manuscript": null, "evidence": [ "lean/docs/299.md:9: No nuclearity, simplicity, trace, or comparison hypothesis is imposed." ] }, { "clause": "Also, the infinite minimal tensor power of every such algebra without characters is Jiang–Su stable, answering the Dadarlat–Toms question.", "status": "undeclared", "coveredBy": [], "manuscript": "The-Kirchberg-Rordam-character-criterion-September-25-2026", "evidence": [ "preprints/The-Kirchberg-Rordam-character-criterion-September-25-2026/build/sections/introduction.tex:70: \\begin{corollary}[Infinite minimal tensor powers]\\label{cor:infinite-tensor}", "preprints/The-Kirchberg-Rordam-character-criterion-September-25-2026/build/sections/introduction.tex:78: Then $B\\cong B\\otmin\\Z$, and $B$ contains a unital copy of $\\Z$.", "lean/docs/299.md:1: # The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability", "lean/docs/299.md:9: The formalized result proves that, for every nonzero unital separable complex $C^*$-algebra $A$ and every free ultrafilter on $\\mathbb N$, the norm central-sequence algebra has no nonzero character exactly when $A\\cong A\\otimes_{\\min}\\mathcal Z$.", "lean/docs/299.md:15: | Kirchberg–Rørdam character criterion | [CharacterCriterion.lean](../ComparatorChallenges/CharacterCriterion.lean) |" ] } ], "gap": "Clause 3 — the infinite minimal tensor power D^{⊗min ∞} of every nonzero unital separable characterless C*-algebra is Z-stable (Dadarlat–Toms question; manuscript corollary cor:infinite-tensor) — is claimed by the headline and by the scope note's title but is not stated by any challenge. Clauses 1–2 (the character criterion) are stated in full.", "scopeNoteOnGap": "silent — the scope note's title names the omitted result (docs/299.md:1: \"# The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability\") while its Scope section describes only the criterion (docs/299.md:9: \"The formalized result proves that, for every nonzero unital separable complex $C^*$-algebra $A$ and every free ultrafilter on $\\mathbb N$, the norm central-sequence algebra has no nonzero character exactly when $A\\cong A\\otimes_{\\min}\\mathcal Z$.\") and the link table lists one row (docs/299.md:15). No sentence says the tensor-power result is not formalized; the omission shows only by enumeration.", "note": "LARGE STATEMENT (2,138 lines, one theorem): I read through the definitions the theorem reaches (norm ultrapower and centralizer F_ω(A) = A_ω ∩ A', HasNoCharacters, MinTensor as the sup over unital representation pairs of the spatial norm, the C*-inductive limit, the prime dimension-drop stages I(2^{7^n}, 2^{7^n}+1) with eigenvalue paths t/2, (t+1)/2 and 1/2, the multiplicities). None of them carries the conclusion. Named encoding: Z is one explicit presentation, and calling it the Jiang–Su algebra rests on Jiang–Su's uniqueness theorem (a simple monotracial limit of prime dimension-drop algebras is Z), which is not formalized; the contracting, dyadic-dense eigenvalue paths make the limit simple and monotracial on an informal reading. The criterion is stated; the per-ω iff implies both readings of 'for every free ultrafilter'. The infinite-tensor-power corollary (Dadarlat–Toms) is stated by no challenge: it is in the linked manuscript, and the note names it only in its heading. Agrees with the first reader.", "firstSuspicious": "Two items for the second reader. (1) The word turns on DECLARED vs UNDECLARED. The scope note's Scope section (docs/299.md:9) and link table (:15) describe only the criterion, so the omission can be inferred by enumeration. But no sentence says the infinite tensor-power result is not formalized, and the note's own title (:1) advertises 'infinite tensor-power Jiang–Su stability'. Under the strict rule this is UNDECLARED. The criterion itself (clauses 1-2) MATCHES with no gap. Clause 3 is a short mathematical corollary of the formalized criterion, but it is neither defined nor stated. (2) The Star instance on A ⊗_min Z. The challenge gives MinTensor.Raw no NormedStarGroup instance; the solution adds one (OAI/Analysis/CharacterCriterion/MinTensor.lean:231) and forces `attribute [local instance 1100] StarCompletion.completionStar` (Main.lean:50) before restating the identical theorem. Mathlib@d13f23b7 has no Star instance on UniformSpace.Completion (checked UniformMulAction, UniformRing, GroupCompletion, Topology/UniformSpace/Completion, Analysis/Normed/*/Completion). The statement therefore elaborates only if completionStar's unused [NormedStarGroup R] section variable (line 411) is dropped, which Lean's rule for definitions should do. If it is not dropped, line 2132 has no Star instance for its target. Lane K's build settles this; by value the involution is correct either way. Also: the solution file InfiniteTensor.lean proves internal infinite-maximal-tensor results (e.g. central_no_character, line 130), but no Comparator challenge states the minimal-tensor-power corollary.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "CharacterCriterion", "flag": "LARGE STATEMENT: 2,138 lines on disk (census 2,139), 145 declarations, one theorem. The definitional closure of character_criterion is essentially the whole file: NormUltrapower (84-405), StarCompletion (407-459), CStarInductiveLimit (461-651), MaxTensor CrossSeminorm/Raw/liftRaw/representationSeminorm (653-725, reached through MinTensor), Spatial (727-882), MinTensor (884-1027), and the Jiang–Su presentation (1029-2121: MatrixTensor, PrimeDimensionDrop, UnitaryPaths, MatrixPermutationPath, MatrixDiagonalPath, MatrixBlockReindex.sumBlock, BalancedDropSystem, GeneralBlockReindex, GeneralDropSystem, PrimeDropPresentation, StandardPrimeMultiplicity, StandardPrimeModel, JiangSu). The only declarations outside it are the section-local instance at line 957 and the empty namespace block at lines 1316-1322; nothing large lies outside. Read through. No definition found to carry the conclusion: Z is a genuine (non-degenerate) Jiang–Su presentation, ⊗_min a genuine spatial norm, F_ω(A) the genuine centralizer. Prop-valued presentation fields (exponent/size_power/zero_copies/one_copies, lines 1930-1934) do not enter the algebra." }, { "challenge": "CharacterCriterion", "flag": "VACUITY?: none found. The hypotheses are jointly satisfiable (e.g. A = ℂ or a UHF algebra, with a non-principal ultrafilter). JiangSu.Algebra is a nonzero unital limit. Unital representations on the zero Hilbert space only contribute 0 to the sup." }, { "challenge": "CharacterCriterion", "flag": "JUNK?: none found. normLimit is .choose of the ω-limit, unique in ℝ (lines 90-100). The Seminorm ⨆ (line 949) is the pointwise sup because the family is bounded above (tmul_le, line 660; bound proved at 979-993). Completion.map star and the Completion SMul are value-correct because star and scalar multiplication are uniformly continuous for the spatial seminorm." }, { "challenge": "CharacterCriterion", "flag": "QUANTIFIER?: matches. ∀ A, ∀ ω free: (no characters on F_ω(A)) ↔ Z-stable (lines 2129-2132), the manuscript's 'let ω be any free ultrafilter' (introduction.tex:38-43). This implies the reading 'Z-stable ↔ for every ω, F_ω(A) has no characters'." }, { "challenge": "CharacterCriterion", "flag": "DEFINITION HOLE: none (definition_names = [], CharacterCriterion.json)." }, { "challenge": "CharacterCriterion", "flag": "OMISSION: headline clause 3 (infinite minimal tensor power, Dadarlat–Toms) is not stated; the scope note's title names it and no sentence says it is not formalized (docs/299.md:1, :9, :15)." } ], "flagReadings": null, "readings": [ "first/batch7.json", "second/second3.json" ] }, { "family": "301", "title": "Trace cones and Razak–Jacelon stabilization", "headline": "Classifies separable nuclear complex C∗-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their full topological cones of extended lower-semicontinuous tracial weights. This answers Robert’s trace-cone question, including algebras with arbitrary ideal structure and both finite and infinite subquotients.", "challenges": [ "TraceIdealTransportSupport" ], "word": "SUPPORT-ONLY", "firstWord": "SUPPORT-ONLY", "secondWord": "SUPPORT-ONLY", "disagreement": null, "clauses": [ { "clause": "Classifies separable nuclear complex C∗-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their full topological cones of extended lower-semicontinuous tracial weights.", "status": "declared", "coveredBy": [], "manuscript": "The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026", "evidence": [ "lean/docs/301.md:11: Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.", "lean/ComparatorChallenges/README.md:23: - `TraceIdealTransportSupport.json`: transport of finite and zero ideals, with" ] }, { "clause": "This answers Robert’s trace-cone question, including algebras with arbitrary ideal structure and both finite and infinite subquotients.", "status": "declared", "coveredBy": [], "manuscript": "The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026", "evidence": [ "lean/docs/301.md:11: Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison." ] } ], "gap": "The classification itself (spatial classification, stabilization models, norm uniqueness, intertwining).", "scopeNoteOnGap": "Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.", "note": "The family's only challenge, TraceIdealTransportSupport, is on the README's list of supporting results, and the note names the classification's missing parts. So SUPPORT-ONLY. Agree with the first reader. The statement imports only Mathlib, and its 17 theorem names are all inside the closure the note credits.", "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch1.json", "second/second4.json" ] }, { "family": "303", "title": "Weak pure infiniteness and Cuntz-algebra absorption", "headline": "Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C∗-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs $`\\mathcal O_\\infty`$, without unitality or simplicity assumptions.", "challenges": [ "ExactInfiniteness", "IndividualStrongInfiniteness" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C∗-algebras.", "status": "stated", "coveredBy": [ "IndividualStrongInfiniteness" ], "reading": "individual_to_strong_main: if every positive h is properly infinite (h⊕h Cuntz-below h in the stabilization), A has the unrestricted diagonalization property and is strongly purely infinite (x², y², xy form). Named encoding: ordinary pure infiniteness is encoded by its Kirchberg–Rørdam characterization 'every positive element is properly infinite' (KR 2000, Theorem 4.16), the form the manuscript itself uses; a reader who refuses that encoding would mark this clause undeclared, which leaves the family word unchanged.", "evidence": [ "lean/ComparatorChallenges/IndividualStrongInfiniteness.lean:316: def PropertyP (A : Type*) [NonUnitalCStarAlgebra A] : Prop :=", "lean/ComparatorChallenges/IndividualStrongInfiniteness.lean:319: theorem individual_to_strong_main (hP : PropertyP A) :", "lean/ComparatorChallenges/IndividualStrongInfiniteness.lean:325: MainGap.StronglyPurelyInfinite A := by", "preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026/build/introduction.tex:46: Condition \\(\\propP\\) is equivalent to pure infiniteness in the", "preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026/build/introduction.tex:48: \\cite[Theorem~4.16]{KR2000}" ] }, { "clause": "For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices.", "status": "stated", "coveredBy": [ "ExactInfiniteness" ], "reading": "Exact A ∧ ∃ n ≥ 1 with diag(a,…,a) (n copies) properly infinite for every nonzero positive a ⇒ every positive a properly infinite and A strongly purely infinite. Encoding: exactness defined from scratch via a hand-built minimal tensor product (C, D in A's universe).", "evidence": [ "lean/ComparatorChallenges/ExactInfiniteness.lean:155: ∃ n : ℕ, 1 ≤ n ∧ ∀ a : A, 0 ≤ a → a ≠ 0 → ProperlyInfinite (diagonalCopies n a)", "lean/ComparatorChallenges/ExactInfiniteness.lean:170: theorem exact_weaklyPurelyInfinite_stronglyPurelyInfinite", "lean/ComparatorChallenges/ExactInfiniteness.lean:172: (hExact : Exact A) (hWeak : WeaklyPurelyInfinite A) : StronglyPurelyInfinite A := by" ] }, { "clause": "Consequently, every separable nuclear algebra with this property absorbs $`\\mathcal O_\\infty`$, without unitality or simplicity assumptions.", "status": "undeclared", "coveredBy": [], "manuscript": "Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026", "classicalTheorem": "Kirchberg–Rørdam (2002) / Kirchberg (2006, Prop. 4.4 and Remark 4.6): a separable nuclear strongly purely infinite algebra absorbs O∞; also nuclear ⇒ exact — not formalized; no challenge mentions O∞, tensor absorption, separability or nuclearity.", "evidence": [ "preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026/build/introduction.tex:129: \\begin{corollary}\\label{cor:absorption}", "preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026/build/introduction.tex:134: A\\cong A\\otimes_{\\min}\\Oinf.", "preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026/build/introduction.tex:125: \\cite[Proposition~4.4 and Remark~4.6]{Kirchberg2006}", "lean/docs/303.md:1: # Weak pure infiniteness and Cuntz-algebra absorption", "lean/docs/303.md:5: - [Weak pure infiniteness and $\\mathcal O_\\infty$ absorption](../../preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026/paper.pdf)", "lean/docs/303.md:11: For exact algebras, proper infiniteness of one fixed finite amplification of every positive element already implies individual proper infiniteness and strong pure infiniteness." ] } ], "gap": "The O_∞-absorption clause (the family title's 'Cuntz-algebra absorption') is not stated; it is a consequence of the stated result via the (unformalized) Kirchberg–Rørdam absorption theorem.", "scopeNoteOnGap": "silent — docs/303.md describes only the two pure-infiniteness implications (lines 9 and 11) and never says the O_∞-absorption consequence is left out; 'absorption' occurs only in its title (line 1) and the linked paper title (line 5).", "note": "Both pure-infiniteness implications are stated. The O∞-absorption consequence (half the family title) is the linked manuscript's Corollary cor:absorption, reached only through Kirchberg–Rørdam/Kirchberg absorption; docs/303.md describes the two implications and never says absorption is left out (the title alone declares nothing): UNDECLARED. Agrees with the first reader.", "firstSuspicious": "The omitted clause is a 'Consequently' corollary through a published theorem (Kirchberg–Rørdam absorption), like 252 (Malcev) and 287 (fundamental group); unlike those, the scope note does not declare it, hence UNDECLARED. A second reader should weigh whether a corollary-by-citation omission belongs in the same bucket as a missing main theorem.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json", "second/second5.json" ] }, { "family": "307", "title": "Failure of rational injectivity for maximal coarse assembly", "headline": "Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class. A companion gives the analogous failure for reduced coarse assembly, disproving the rational coarse Novikov conjecture.", "challenges": [ "CoarseAssembly" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective.", "status": "unlinked", "coveredBy": [], "manuscript": "Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026", "evidence": [ "preprints/Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026/build/main.tex:9-13: We construct a uniformly discrete bounded-geometry space whose maximal coarse ⏎ assembly map has an infinite-order element in its kernel. The space is a coarse ⏎ disjoint union of finite connected graphs of uniformly bounded degree, so ⏎ maximal coarse assembly need not be rationally injective even for such graph ⏎ unions.", "preprints/Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026/build/sections/01-introduction.tex:36-44: \\begin{theorem}\\label{thm:main} ⏎ There exist finite connected graphs $(X_j)_{j\\geq1}$ with a common finite ⏎ bound on vertex degree, a coarse disjoint union $X=\\bigsqcup_{j\\geq1}X_j$, ⏎ and an infinite-order class $\\alpha\\in\\KX_1(X)$ such that ⏎ \\[ ⏎ \\mu_X^{\\max}(\\alpha)=0. ⏎ \\] ⏎ Consequently, $\\mu_X^{\\max}\\otimes\\Q$ is not injective. ⏎ \\end{theorem}", "preprints/Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026/build/sections/01-introduction.tex:46-53: The theorem strengthens the reduced coarse-assembly result proved in the ⏎ companion paper~\\cite{OpenAICoarseNovikov2026}. ⏎ If $C^*(X)$ denotes the operator-norm ⏎ closure of $\\C[X]$, the canonical quotient ⏎ $C^*_{\\max}(X)\\to C^*(X)$ carries the maximal index to the reduced index. ⏎ Our class $\\alpha$ therefore also lies in the reduced assembly kernel. The stronger ⏎ conclusion requires a proof in the universal norm: vanishing of a reduced ⏎ index alone does not imply vanishing of its maximal lift.", "lean/docs/307.md:1: # Failure of rational injectivity for maximal coarse assembly", "lean/docs/307.md:5: - [A counterexample to the coarse Novikov conjecture](../../preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/paper.pdf)", "lean/docs/307.md:9: The coarse Novikov conjecture predicts rational injectivity of the ordinary coarse assembly map for uniformly discrete spaces of bounded geometry. The formalization constructs a counterexample from a coarse disjoint union of finite connected graphs with uniformly bounded degree. Its degree-one coarse $K$-homology contains an infinite-order class whose image under ordinary coarse assembly into the reduced locally compact Roe algebra vanishes. The class remains nonzero after tensoring with $\\mathbb Q$, so rational injectivity fails.", "lean/docs/307.md:15: | Failure of rational injectivity for ordinary coarse assembly | [CoarseAssembly.lean](../ComparatorChallenges/CoarseAssembly.lean) |", "preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/01-introduction.tex:91-93: extension. Throughout, our target is the reduced, locally compact Roe algebra; ⏎ maximal completions and coefficient-free group assembly are different ⏎ questions.", "lean/ComparatorChallenges/CoarseAssembly.lean:10: ordinary reduced coarse assembly map, including after rationalization.", "lean/ComparatorChallenges/CoarseAssembly.lean:626-627: def algebra : NonUnitalStarSubalgebra ℂ (Operator X H) := ⏎ (NonUnitalStarAlgebra.adjoin ℂ {T : Operator X H | Generator T}).topologicalClosure", "lean/ComparatorChallenges/CoarseAssembly.lean:6489: def TotalRoe : Type := algebra (X := VertexUnion) (H := AmpleSpace)", "lean/ComparatorChallenges/CoarseAssembly.lean:6610-6611: def constructedCoarseAssembly : KX1 VertexUnion →+ OperatorKTheory.K1 TotalRoe := ⏎ constructedTotalIndex.assembly", "lean/ComparatorChallenges/CoarseAssembly.lean:6636-6640: ∃ α : KX1 VertexUnion, ⏎ (∀ n : ℤ, n ≠ 0 → n • α ≠ 0) ∧ constructedCoarseAssembly α = 0 ∧ ⏎ (1 : ℚ) ⊗ₜ[ℤ] α ≠ 0 ∧ ⏎ ¬ Function.Injective constructedCoarseAssembly ∧ ⏎ ¬ Function.Injective (constructedCoarseAssembly.toIntLinearMap.lTensor ℚ) := by" ] }, { "clause": "The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class.", "status": "unlinked", "coveredBy": [], "manuscript": "Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026", "partial": "the same shape — a coarse disjoint union of finite connected graphs of bounded degree with an infinite-order class — is stated, but for the September-23 construction (discs in the triangular lattice) and with the class in the REDUCED kernel; \"the example\" here is the October-5 construction (three-dimensional discrete tori) and its class lies in the maximal kernel", "evidence": [ "preprints/Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026/build/sections/01-introduction.tex:125-126: The base graphs are three-dimensional discrete tori $B_j$ whose side ⏎ lengths tend to infinity. Over each $B_j$ we place a finite quotient of", "lean/ComparatorChallenges/CoarseAssembly.lean:716: def triangularGraph : SimpleGraph LatticeVertex where", "lean/ComparatorChallenges/CoarseAssembly.lean:722: def Disc (j : ℕ) := {u : LatticeVertex // latticeRadius u ≤ (j : ℤ)}", "preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/01-introduction.tex:101: Begin with finite discs $D_j$ of radii $j$ in the triangular lattice, and let $C_j$ be their", "lean/docs/307.md:5: - [A counterexample to the coarse Novikov conjecture](../../preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/paper.pdf)" ] }, { "clause": "A companion gives the analogous failure for reduced coarse assembly, disproving the rational coarse Novikov conjecture.", "status": "stated", "coveredBy": [ "CoarseAssembly" ], "encoding": "reduced Roe algebra = operator-norm closure in B(ℓ²(X, ℓ²ℕ)) of the *-algebra generated by finite-propagation, locally compact operators; KX1 = direct limit over Rips scales of a bespoke odd analytic K-homology (Fredholm-module cycles modulo degenerate / unitary-equivalence / direct-sum / homotopy relations); K1 a bespoke presentation; assembly a bespoke boundary index transported from the separated fibre union; uniformly discrete as 1 ≤ dist, bounded geometry as UniformFiniteBalls", "evidence": [ "lean/ComparatorChallenges/CoarseAssembly.lean:6629-6640: theorem bounded_geometry_graph_counterexample_unconditional : ⏎ (∀ j, Finite (FiberVertex j) ∧ ⏎ (fiberedGraph elementaryMoves chosenHolonomyUnit j).Connected) ∧ ⏎ (∃ D : ℕ, ∀ j (x : FiberVertex j), ⏎ Nat.card {y // (fiberedGraph elementaryMoves chosenHolonomyUnit j).Adj x y} ≤ D) ∧ ⏎ ProperSpace VertexUnion ∧ UniformFiniteBalls VertexUnion ∧ ⏎ (∀ x y : VertexUnion, x ≠ y → 1 ≤ dist x y) ∧ ⏎ ∃ α : KX1 VertexUnion, ⏎ (∀ n : ℤ, n ≠ 0 → n • α ≠ 0) ∧ constructedCoarseAssembly α = 0 ∧ ⏎ (1 : ℚ) ⊗ₜ[ℤ] α ≠ 0 ∧ ⏎ ¬ Function.Injective constructedCoarseAssembly ∧ ⏎ ¬ Function.Injective (constructedCoarseAssembly.toIntLinearMap.lTensor ℚ) := by", "preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/01-introduction.tex:49-58: \\begin{theorem}\\label{thm:main} ⏎ There is a uniformly discrete bounded-geometry metric space $X$, which is a ⏎ coarse disjoint union of finite connected graphs of uniformly bounded degree, ⏎ and an infinite-order element $\\alpha\\in KX_1(X)$ such that ⏎ \\[ ⏎ \\mu_X(\\alpha)=0\\quad\\text{in }K_1(C^*(X)). ⏎ \\] ⏎ Consequently, ordinary coarse assembly for $X$ fails to be injective both ⏎ integrally and after tensoring its domain and target with $\\Q$. ⏎ \\end{theorem}" ] } ], "gap": "failure of rational injectivity for the MAXIMAL coarse assembly map (the headline's lead claim, the October-5 manuscript); only the reduced (September-23 companion) result is stated", "scopeNoteOnGap": "silent on the maximal clause — docs/307.md is titled \"# Failure of rational injectivity for maximal coarse assembly\" (line 1), yet links only the September-23 reduced paper (line 5) and describes only \"ordinary coarse assembly into the reduced locally compact Roe algebra\" (line 9); it never says the maximal statement is not formalized", "note": "Read with care: the Lean target is K1 of the REDUCED Roe algebra (TotalRoe := DiscreteRoe.algebra, an operator-norm closure inside B(ℓ²); docstring line 10 \"ordinary reduced coarse assembly map\"), and the space is the September-23 disc construction. The maximal statement is not stated and not implied — the October-5 paper says itself \"vanishing of a reduced index alone does not imply vanishing of its maximal lift\" (01-introduction.tex:52-53), and its example is a different space. The maximal clauses come only from the October-5 paper, which the note does not link; the linked paper calls maximal completions \"different questions\" (01-introduction.tex:91-93). Hence UNLINKED, not UNDECLARED (first reader, before amendment 3). One thing a reader should see: the note's TITLE (307.md:1) is \"Failure of rational injectivity for maximal coarse assembly\" while its body (307.md:9) and its comparator label (307.md:15) say reduced/ordinary — a title alone neither declares nor states, but it is the one place in the release where the formalization page names the unformalized maximal result as its subject.", "firstSuspicious": "Borderline DECLARED/UNDECLARED: the scope note's body names the reduced target, but its own title claims 'maximal' and it never says the maximal theorem is left out. Also re-check the two guarded `else 0` fallbacks. Neither coarse-assembly preprint appears in lean/formalization.yaml (per the delegated reader).", "confidence": "medium", "conditionalOnLaneK": null, "flags": [ { "challenge": "CoarseAssembly", "flag": "NARROWER: the Roe algebra is reduced, not maximal (CoarseAssembly.lean:626-627 operator-norm closure in B(ℓ²); docstring line 10 'ordinary reduced coarse assembly map')" }, { "challenge": "CoarseAssembly", "flag": "DEFINITION CARRIES? (guarded fallback to zero): analyticFamilyIndex (5783-5785) and separatedIndexFamily (6226-6230) are defined by `if then else 0`; if either guard were false, constructedCoarseAssembly would be identically 0, 'assembly α = 0' and non-injectivity would be free, and the theorem would reduce to 'KX1 has an infinite-order element'. The statement alone does not certify the guards; the delegated closure read reports solution-tree lemmas proving both (OAI/Topology/CoarseAssembly/AnalyticFamilyIndex.lean:22-60, SeparatedCoarseAssembly.lean:22-32), read but not run — lane K should confirm" }, { "challenge": "CoarseAssembly", "flag": "LARGE STATEMENT: 6,649 lines; the whole declaration closure (217 declarations) was read by a delegated reader; no other degenerate definition found; chosenHolonomyUnit is Classical.choose but its spec is carried in the definition" } ], "flagReadings": null, "readings": [ "first/batch8.json", "second/second6.json" ] }, { "family": "312", "title": "The Grothendieck homotopy hypothesis", "headline": "Proves the Grothendieck homotopy hypothesis for ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces.", "challenges": [ "GrothendieckElementaryExpansion" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves the Grothendieck homotopy hypothesis for ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces.", "coveredBy": [ "GrothendieckElementaryExpansion" ], "note": "Only an ingredient is stated: for every Grothendieck coherator (Ara–Henry convention), every cellular model X, and every pushout of the source inclusion D_n → D_{n+1} along a map D_n → X, the induced X → Y is a weak equivalence (bijection on components and on all based homotopy sets). The homotopy hypothesis itself (equivalence of homotopy theories with spaces) is not stated." } ], "gap": "The Grothendieck homotopy hypothesis (the comparison of the homotopy theory of coherator ∞-groupoids with spaces) is not stated; only the elementary-expansion theorem (Henry's pushout conjecture) is.", "scopeNoteOnGap": "The later semi-model structure and full comparison with the homotopy theory of spaces are not included.", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "GrothendieckElementaryExpansion", "flag": "VACUITY? (checked, not found): IsCoherator, Model, IsCellular are all inhabited by the standard objects (Θ₀ tower, initial model); no hypothesis appears contradictory." } ], "flagReadings": [ { "challenge": "GrothendieckElementaryExpansion", "flag": "VACUITY? (checked, not found): IsCoherator, Model, IsCellular are all inhabited by the standard objects (Θ₀ tower, initial model); no hypothesis appears contradictory.", "decision": "cleared", "why": "All hypotheses can be met at once. (1) A coherator exists. stage 0 can be Θ₀ (Hom S T := ULift (S.realize ⟶ T.realize), base = up). It is initial, because map_base forces every Morphism out of it, and it preserves globular sums, because Shape.attach is realized as the presheaf pushout. Free extensions along admissible pairs and sequential colimits exist among globular theories (an essentially algebraic, locally presentable category). Adjoining all admissible pairs at every stage gives the contractibility conjunct of IsCoherator. (2) Models exist: the constant-PUnit presheaf satisfies the pullback condition, and freeShape / disk are models by construction (lines 343-348). (3) For the remaining hypotheses take X = disk C n, a = 𝟙, Y = disk C (n+1), i = J C n, b = 𝟙. IsPushout 𝟙 J J 𝟙 holds trivially, and disk C n is cellular: the free model on the n-globe is built from the empty initial model by attaching its 2 cells per dimension below n and one n-cell. This instance has content, since it asserts that J is a weak equivalence. Restricting admissible pairs to height ≤ n+1 (line 271) only removes filler obligations. That enlarges the class of coherators and makes the theorem stronger, not vacuous.", "evidence": [ "lean/ComparatorChallenges/GrothendieckElementaryExpansion.lean:271: height : arity.height ≤ n + 1", "lean/ComparatorChallenges/GrothendieckElementaryExpansion.lean:292: initial : ∀ D : GlobularTheory.{v}, Nonempty (Unique (Morphism (stage 0) D))", "lean/ComparatorChallenges/GrothendieckElementaryExpansion.lean:304: def IsCoherator (C : GlobularTheory.{v}) : Prop :=", "lean/ComparatorChallenges/GrothendieckElementaryExpansion.lean:305: Nonempty C.CellularPresentation ∧", "lean/ComparatorChallenges/GrothendieckElementaryExpansion.lean:306: ∀ p : C.AdmissiblePair, ∃ z, Fills (Morphism.id C) p z", "lean/ComparatorChallenges/GrothendieckElementaryExpansion.lean:344: presheaf := yoneda.obj ⟨S⟩", "lean/ComparatorChallenges/GrothendieckElementaryExpansion.lean:350: noncomputable def J (C : GlobularTheory.{v}) (n : ℕ) : disk C n ⟶ disk C (n + 1) :=", "lean/ComparatorChallenges/GrothendieckElementaryExpansion.lean:352: yoneda.map (C.toTheoryData.arity.map (Shape.source n))", "lean/ComparatorChallenges/GrothendieckElementaryExpansion.lean:380: def IsCellular {C : GlobularTheory.{v}} (X : C.Model) : Prop := Nonempty X.CellularPresentation", "lean/ComparatorChallenges/GrothendieckElementaryExpansion.lean:470: (square : IsPushout a (GlobularTheory.Model.J C n) i b) :" ] } ], "readings": [ "first/batch3.json", "flags/flags2.json" ] }, { "family": "317", "title": "Thomason model structures in all strict higher dimensions", "headline": "Resolves the Ara–Maltsiniotis conjecture: for every n ≥ 1 and n = ω, small strict globular n-categories admit proper combinatorial Thomason model structures Quillen equivalent to simplicial sets. Thus strict higher categories model the homotopy theory of spaces in every stated dimension.", "challenges": [ "ThomasonModelStructures" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "for every n ≥ 1 and n = ω, small strict globular n-categories admit proper combinatorial Thomason model structures Quillen equivalent to simplicial sets", "coveredBy": [ "ThomasonModelStructures" ], "note": "FullMain: a Kan–Quillen model on SSet and an ExactModelEndpoint for ω and for every finite n ≥ 1." }, { "clause": "Thus strict higher categories model the homotopy theory of spaces in every stated dimension.", "coveredBy": [ "ThomasonModelStructures" ], "note": "Consequence of the Quillen equivalences." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "medium (2,474-line statement; Mathlib's simplicial layer taken on trust)", "conditionalOnLaneK": null, "flags": [ { "challenge": "ThomasonModelStructures", "flag": "LARGE STATEMENT: 2,474 lines. Closure read: OmegaCategory/Functor/NCategory (12–119), DirectedComplex/Table/nu (129–503), NormalizedChains complex (575–830), simplexComplexes/orientals/ulift/raisedOrientals (1176–1345), Presentation machinery for colimits and truncation (1351–1871, 1983–2300), categorification/streetNerve/thomasonLeft/Right/adjunctions (1873–2323), WeakHomotopy and the endpoint structures (2326–2435). Data-carrying definitions are the standard Steiner ν / Street orientals and nerve; the colimit and truncation constructions enter only through genuine adjunction/colimit structures, so they are fixed up to isomorphism. No degenerate definition found; the Mathlib SSet layer was not re-read." } ], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "322", "title": "Tingley’s sphere-isometry problem", "headline": "Resolves Tingley's problem: every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces. No dimension restriction is imposed, so the metric geometry of the unit sphere determines the Banach space up to linear isometry.", "challenges": [ "TingleySphereIsometry" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves Tingley's problem: every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces.", "coveredBy": [ "TingleySphereIsometry" ], "note": "Complete nontrivial real normed X, Y; surjective isometry f of unit spheres ⇒ ∃ linear isometric equivalence T extending f, with the radial formula, unique among all real-linear extensions." }, { "clause": "No dimension restriction is imposed, so the metric geometry of the unit sphere determines the Banach space up to linear isometry.", "coveredBy": [ "TingleySphereIsometry" ], "note": "No finite-dimensionality or other hypothesis." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch5.json" ] }, { "family": "323", "title": "Independence of the separable quotient problem", "headline": "Establishes, relative to the consistency of a measurable cardinal, that the separable quotient problem is independent of ZFC. The assertion that every infinite-dimensional Banach space has a separable infinite-dimensional quotient can hold for all real and complex Banach spaces, whereas the continuum hypothesis yields counterexamples over both fields.", "challenges": [ "SeparableQuotientNegative" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Establishes, relative to the consistency of a measurable cardinal, that the separable quotient problem is independent of ZFC.", "coveredBy": [ "SeparableQuotientNegative" ], "note": "Only the CH-counterexample half of independence." }, { "clause": "The assertion that every infinite-dimensional Banach space has a separable infinite-dimensional quotient can hold for all real and complex Banach spaces,", "coveredBy": [], "note": "Not stated (the consistency direction from a measurable cardinal)." }, { "clause": "whereas the continuum hypothesis yields counterexamples over both fields.", "coveredBy": [ "SeparableQuotientNegative" ], "note": "CH → ¬SQ ℝ ∧ ¬SQ ℂ, at every universe level." } ], "gap": "The positive consistency direction (from a measurable cardinal), hence the independence statement as a whole.", "scopeNoteOnGap": "lean/docs/323.md:11: \"This is the conditional counterexample direction. The positive consistency direction and the paper's full relative-independence conclusions are outside the selected statement.\"", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch6.json" ] }, { "family": "324", "title": "Lipschitz equivalent Banach spaces need not be linearly isomorphic", "headline": "Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively. Thus even the complete metric structure up to bi-Lipschitz equivalence does not determine a separable Banach space's linear isomorphism class.", "challenges": [ "LipschitzEquivalence", "C0Absorption" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively.", "coveredBy": [ "LipschitzEquivalence" ], "note": "explicit Lipschitz constants 4/21 and 76/25; IsEmpty (X ≃L[ℝ] Y); X ⊇ isometric c0(ℓ2), Y has no bounded-below linear copy of c0(ℓ2)" }, { "clause": "Thus even the complete metric structure up to bi-Lipschitz equivalence does not determine a separable Banach space's linear isomorphism class.", "coveredBy": [ "LipschitzEquivalence" ], "note": "restatement" } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "LipschitzEquivalence", "flag": "VACUITY? cleared: the isometric copy of c0(ℓ2) in X makes X nontrivial, so IsEmpty (X ≃L Y) is not satisfied trivially by zero spaces" } ], "flagReadings": [ { "challenge": "LipschitzEquivalence", "flag": "VACUITY? cleared: the isometric copy of c0(ℓ2) in X makes X nontrivial, so IsEmpty (X ≃L Y) is not satisfied trivially by zero spaces", "decision": "cleared", "why": "IsEmpty (X ≃L Y) could be cheap only for degenerate spaces. X contains a linear isometric copy of c0(ℓ2), which is infinite-dimensional, and Ψ is a bijection X ≃ Y with two-sided Lipschitz bounds, so Y is nonzero and of the same cardinality. Non-isomorphism is then the genuine content, and it also follows honestly from the last two conjuncts: an isomorphism would carry the c0(ℓ2) copy into Y, contradicting ¬ContainsLinearCopy. ContainsLinearCopy is the genuine bounded-below linear embedding. Lane K: CERTIFIED.", "evidence": [ "ComparatorChallenges/LipschitzEquivalence.lean:24: def ContainsLinearCopy (E : Type uE) (F : Type uF) [NormedAddCommGroup E] [NormedSpace ℝ E]", "ComparatorChallenges/LipschitzEquivalence.lean:25: [NormedAddCommGroup F] [NormedSpace ℝ F] : Prop :=", "ComparatorChallenges/LipschitzEquivalence.lean:26: ∃ (T : E →L[ℝ] F) (a : ℝ), 0 < a ∧ ∀ x : E, a * ‖x‖ ≤ ‖T x‖", "ComparatorChallenges/LipschitzEquivalence.lean:28: def MainClaim : Prop :=", "ComparatorChallenges/LipschitzEquivalence.lean:29: ∃ X Y : SeparableRealBanach,", "ComparatorChallenges/LipschitzEquivalence.lean:30: ∃ Ψ : X.Carrier ≃ Y.Carrier,", "ComparatorChallenges/LipschitzEquivalence.lean:31: (∀ s t : X.Carrier,", "ComparatorChallenges/LipschitzEquivalence.lean:32: (4 / 21 : ℝ) * ‖s - t‖ ≤ ‖Ψ s - Ψ t‖ ∧", "ComparatorChallenges/LipschitzEquivalence.lean:33: ‖Ψ s - Ψ t‖ ≤ (76 / 25 : ℝ) * ‖s - t‖) ∧", "ComparatorChallenges/LipschitzEquivalence.lean:34: IsEmpty (X.Carrier ≃L[ℝ] Y.Carrier) ∧", "ComparatorChallenges/LipschitzEquivalence.lean:35: Nonempty (C0L2 →ₗᵢ[ℝ] X.Carrier) ∧", "ComparatorChallenges/LipschitzEquivalence.lean:36: ¬ ContainsLinearCopy C0L2 Y.Carrier" ] } ], "readings": [ "first/batch8.json", "flags/flags3.json" ] }, { "family": "325", "title": "The complete Crouzeix conjecture", "headline": "Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has $`\\lVert P[A]\\rVert\\le2\\sup_{z\\in W(A)}\\lVert P(z)\\rVert`$, where $`W(A)`$ is the numerical range. The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions.", "challenges": [ "DirectCrouzeix", "CompleteCrouzeix", "StructuralCrouzeix", "CrouzeixHilbert", "HilbertCrouzeix" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has $`\\lVert P[A]\\rVert\\le2\\sup_{z\\in W(A)}\\lVert P(z)\\rVert`$, where $`W(A)`$ is the numerical range", "coveredBy": [ "CrouzeixHilbert", "DirectCrouzeix", "CompleteCrouzeix" ], "note": "CrouzeixHilbert: every complete complex inner-product space H (any universe, no separability), every bounded A, every m > 0 and matrix polynomial; H = 0 handled separately. The matrix files give the finite-dimensional case." }, { "clause": "The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions.", "coveredBy": [ "DirectCrouzeix", "CompleteCrouzeix", "CrouzeixHilbert" ], "note": "uniform_sharpness / CompleteCrouzeix.main: no universal constant below 2; SharpConstant: a 2×2 example with sup 1 and norm 2." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "326", "title": "The cotype–cotype conjecture under the approximation property", "headline": "Resolves the cotype–cotype conjecture for real Banach spaces with the approximation property. Such a nonzero space is K-convex if and only if both it and its dual have finite Rademacher cotype, with possibly different exponents. Equivalently, these cotype assumptions force nontrivial Rademacher type.", "challenges": [ "Cotype" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Resolves the cotype–cotype conjecture for real Banach spaces with the approximation property.", "status": "stated", "coveredBy": [ "Cotype" ], "evidence": [ "lean/ComparatorChallenges/Cotype.lean:47: def MainTarget [CompleteSpace E] [Nontrivial E] : Prop :=", "lean/ComparatorChallenges/Cotype.lean:53: theorem mainTarget_proved [Nontrivial X] : MainTarget X := by" ] }, { "clause": "Such a nonzero space is K-convex if and only if both it and its dual have finite Rademacher cotype, with possibly different exponents.", "status": "stated", "coveredBy": [ "Cotype" ], "evidence": [ "lean/ComparatorChallenges/Cotype.lean:48: ApproximationProperty E →", "lean/ComparatorChallenges/Cotype.lean:49: (KConvex E ↔ FiniteCotype E ∧ FiniteCotype (E →L[ℝ] ℝ))", "lean/ComparatorChallenges/Cotype.lean:40: def FiniteCotype : Prop := ∃ q : ℝ, HasCotype E q" ], "note": "Each side has its own ∃ q, so the exponents may differ; K-convexity as a uniform bound on the Rademacher projection over finite cubes; real scalars; Nontrivial = nonzero." }, { "clause": "Equivalently, these cotype assumptions force nontrivial Rademacher type.", "status": "undeclared", "coveredBy": [], "manuscript": "The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026", "evidence": [ "preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/build/introduction.tex:65: The classical theorems of Maurey--Pisier and Pisier", "preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/build/introduction.tex:66: \\cite{MP76,Pis82K} identify \\(K\\)-convexity with nontrivial type:", "preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/build/introduction.tex:71: X\\text{ has Rademacher type }p\\text{ for some }p>1\\\\", "lean/docs/326.md:5: - [The cotype–cotype conjecture under the approximation property](../../preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/paper.pdf)", "lean/docs/326.md:9: The formalized result establishes this equivalence for every nonzero real Banach space with the approximation property: $X$ is $K$-convex exactly when both $X$ and $X^*$ have finite cotype." ], "note": "Rademacher type is not defined in the challenge; the clause follows from the stated equivalence only through Pisier's K-convexity theorem (Maurey–Pisier 1976, Pisier 1982: K-convex ⇔ nontrivial type), which is not formalized. The manuscript is linked and the note does not mention type." } ], "gap": "Only the 'Equivalently … nontrivial Rademacher type' sentence, a reformulation of the stated equivalence through Pisier's K-convexity theorem; no notion of type appears in the challenge.", "scopeNoteOnGap": "silent (326.md:9 states only the K-convexity equivalence; the abstract does not mention type either)", "note": "The cotype–cotype equivalence is stated exactly; only the \"Equivalently … nontrivial Rademacher type\" reformulation is missing, and it needs Pisier's unformalized classical theorem: NARROWER — UNDECLARED under the brief's rule (a clause reached only through an unformalized classical theorem is not stated). Agree with the first reader. The gap is a named classical bridge, not new mathematics.", "firstSuspicious": "Word depends on the same classical-bridge question as 089 (here the headline itself says 'Equivalently'). If a clause derivable by one named classical theorem counts as stated, this family is MATCHES.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch6.json", "second/second7.json" ] }, { "family": "327", "title": "Markov type characterizes superreflexivity", "headline": "Proves that every real Banach space with Markov type p for some p > 1 admits an equivalent uniformly convex norm, answering Naor's renorming question. Together with the known converse, this characterizes superreflexivity by nontrivial Markov type.", "challenges": [ "MarkovType" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that every real Banach space with Markov type p for some p > 1 admits an equivalent uniformly convex norm, answering Naor's renorming question.", "coveredBy": [ "MarkovType" ], "note": "Forward direction of the iff." }, { "clause": "Together with the known converse, this characterizes superreflexivity by nontrivial Markov type.", "coveredBy": [ "MarkovType" ], "note": "The iff includes the converse; superreflexivity is encoded as admitting an equivalent uniformly convex norm (Enflo–Pisier)." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch5.json" ] }, { "family": "328", "title": "Nonexpansive fixed points in reflexive Banach spaces", "headline": "Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without assuming uniform convexity.", "challenges": [ "ReflexiveFixedPoints" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point", "coveredBy": [ "ReflexiveFixedPoints" ], "note": "For every real Banach space X (CompleteSpace) whose canonical map into the continuous bidual is surjective, every nonempty closed bounded convex C ⊆ X and every F : C → C with ‖F a − F b‖ ≤ ‖a − b‖, ∃ a, F a = a." }, { "clause": "The result uses the original norm, without assuming uniform convexity", "coveredBy": [ "ReflexiveFixedPoints" ], "note": "Nonexpansiveness is in X's given norm; no renorming, uniform convexity, normal structure or separability hypothesis." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch2.json" ] }, { "family": "329", "title": "A counterexample to metric-entropy duality", "headline": "Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube.", "challenges": [ "MetricEntropyDuality" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Disproves Pietsch's dimension-free duality conjecture for metric entropy.", "coveredBy": [ "MetricEntropyDuality" ], "note": "∀ a, b ≥ 1 a counterexample exists." }, { "clause": "Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube.", "coveredBy": [ "MetricEntropyDuality" ], "note": "b·log N(B∞°, a⁻¹K°) < log N(K, B∞) with K a symmetric convex body in some R^n." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "330", "title": "A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces", "headline": "Constructs a countable uniformly discrete metric space whose real Lipschitz-free Banach space has the approximation property but not the bounded approximation property, answering Kalton's question negatively. Finite-rank operators approximate the identity on every compact set, but their norms cannot share a finite bound.", "challenges": [ "RealL1Renorming", "DiscreteLipschitzFree" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a countable uniformly discrete metric space whose real Lipschitz-free Banach space has the approximation property but not the bounded approximation property, answering Kalton's question negatively.", "coveredBy": [ "DiscreteLipschitzFree" ], "note": "∃ countable M with d(x,y) ≥ 1 for x ≠ y (also unbounded and not proper) such that F(M) has AP and ¬BAP(Λ) for every Λ ≥ 1 (Λ < 1 is implied)." }, { "clause": "Finite-rank operators approximate the identity on every compact set, but their norms cannot share a finite bound.", "coveredBy": [ "DiscreteLipschitzFree" ], "note": "HasAP / ∀ Λ, ¬HasBAP Λ." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch6.json" ] }, { "family": "331", "title": "Reflexive midpoint convexity and diamond distortion", "headline": "Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces. In the same space, depth-k countably branching diamonds require distortion at least $`\\sqrt{1+k/12}`$, so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity.", "challenges": [ "ForestSpace", "MidpointLenses", "DiamondDistortion", "DaugavetModuli", "BoundedTreePotentials", "IndependentProducts", "RecursivePotentials" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces.", "coveredBy": [ "ForestSpace" ], "note": "Infinite-dimensional, separable, reflexive full dual of the segment-norm completion; averaged midpoint modulus ≥ √(1+t²/12) − 1 for all t>0 (implies AMUC); no equivalent norm is AUC." }, { "clause": "In the same space, depth-k countably branching diamonds require distortion at least $`\\sqrt{1+k/12}`$,", "coveredBy": [ "ForestSpace" ], "note": "Last conjunct: every embedding with distortion ≤ C (any positive scale) has C ≥ √(1+k/12)." }, { "clause": "so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity.", "coveredBy": [ "ForestSpace" ], "note": "Immediate from the two previous clauses." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": "The headline rests on ForestSpace alone (read in full); the other six challenges are companion results from six further manuscripts and were read for vacuity/junk only. RecursivePotentials' existential over Prop-valued instance classes was checked and is not a definition hole.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [ { "challenge": "ForestSpace", "flag": "JUNK? (not triggered): moduli are nested real sInf/sSup; in an infinite-dimensional space with an equivalent norm the inner sets are nonempty and lie in [−1,t] (AUC) or [0,t] (averaged), so no junk value can occur" }, { "challenge": "MidpointLenses", "flag": "JUNK? (not triggered): Real.sqrt of a negative is 0, but ‖x‖ ≤ (‖x+y‖+‖x−y‖)/2 ≤ R" } ], "flagReadings": [ { "challenge": "ForestSpace", "flag": "JUNK? (not triggered): moduli are nested real sInf/sSup; in an infinite-dimensional space with an equivalent norm the inner sets are nonempty and lie in [−1,t] (AUC) or [0,t] (averaged), so no junk value can occur", "decision": "cleared", "why": "The direction that matters is ¬IsAUC: a junk 0 in aucModulus would make it trivially true. It cannot occur. Conjunct 1 asserts FullDual Vertex is infinite-dimensional, so every closed finite-codimensional F is infinite-dimensional, hence nonzero. N is a norm (α‖x‖ ≤ N x with α > 0), so F contains N-unit vectors and aucValues/averageValues are nonempty. For N x = N y = 1 and t > 0: N(x+ty)−1 ∈ [−1, t] (it is in fact ≥ |t−1|−1), and (N(x+ty)+N(x−ty))/2 − 1 ∈ [0, t] by the triangle inequality. So each inner sInf is a true infimum. The outer range is nonempty (F = the whole space is in Cofinite) and bounded above by t, so the sSup is true. The outermost set is nonempty (unit vectors exist) and bounded below (−1 or 0), so the sInf is true. ¬IsAUC therefore asserts a genuinely nonpositive modulus at some t > 0, and the averaged lower bound √(1+t²/12)−1 > 0 is compared with the true modulus.", "evidence": [ "ComparatorChallenges/ForestSpace.lean:19-20: def Cofinite : Type := ⏎ {F : Submodule ℝ E // IsClosed (F : Set E) ∧ FiniteDimensional ℝ (E ⧸ F)}", "ComparatorChallenges/ForestSpace.lean:23-24: def aucValues (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : Set ℝ := ⏎ {a | ∃ y : E, y ∈ F.val ∧ N y = 1 ∧ a = N (x + t • y) - 1}", "ComparatorChallenges/ForestSpace.lean:26-35: noncomputable def aucInner (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : ℝ := ⏎ sInf (aucValues E N t x F) ⏎ ⏎ noncomputable def aucCenter (N : E → ℝ) (t : ℝ) (x : E) : ℝ := ⏎ sSup (Set.range (aucInner E N t x)) ⏎ ⏎ /-- The one-sided asymptotic uniform convexity modulus, with the order ⏎ of infimum, supremum and infimum. -/ ⏎ noncomputable def aucModulus (N : E → ℝ) (t : ℝ) : ℝ := ⏎ sInf {a | ∃ x : E, N x = 1 ∧ a = aucCenter E N t x}", "ComparatorChallenges/ForestSpace.lean:38-50: def averageValues (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : Set ℝ := ⏎ {a | ∃ y : E, y ∈ F.val ∧ N y = 1 ∧ ⏎ a = (N (x+t•y) + N (x-t•y))/((2 : ℕ) : ℝ) - 1} ⏎ ⏎ noncomputable def averageInner (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : ℝ := ⏎ sInf (averageValues E N t x F) ⏎ ⏎ noncomputable def averageCenter (N : E → ℝ) (t : ℝ) (x : E) : ℝ := ⏎ sSup (Set.range (averageInner E N t x)) ⏎ ⏎ /-- The averaged asymptotic midpoint modulus. -/ ⏎ noncomputable def averageModulus (N : E → ℝ) (t : ℝ) : ℝ := ⏎ sInf {a | ∃ x : E, N x = 1 ∧ a = averageCenter E N t x}", "ComparatorChallenges/ForestSpace.lean:61-62: lower : ∀ x : E, α*‖x‖ ≤ N x ⏎ upper : ∀ x : E, N x ≤ β*‖x‖", "ComparatorChallenges/ForestSpace.lean:69-70: def IsAUC {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] (N : E → ℝ) : Prop := ⏎ ∀ t : ℝ, 0 < t → 0 < aucModulus E N t", "ComparatorChallenges/ForestSpace.lean:454: (¬FiniteDimensional ℝ (FullDual Vertex)) ∧", "ComparatorChallenges/ForestSpace.lean:457-459: (∀ (α β : ℝ) (R : EquivalentNorm (FullDual Vertex) α β), ¬IsAUC R.N) ∧ ⏎ (∀ t : ℝ, 0 < t → ⏎ Real.sqrt (1+t^2/12)-1 ≤ averageModulus (FullDual Vertex) norm t) ∧" ] }, { "challenge": "MidpointLenses", "flag": "JUNK? (not triggered): Real.sqrt of a negative is 0, but ‖x‖ ≤ (‖x+y‖+‖x−y‖)/2 ≤ R", "decision": "cleared", "why": "Real.sqrt of a negative would be 0 and would turn the conclusion into the stronger ‖tail H y‖ ≤ 0, i.e. falsity, never weakness. It never happens: 2x = (x+y)+(x−y) gives ‖x‖ ≤ (‖x+y‖+‖x−y‖)/2 ≤ R and R ≥ 0, so R² − ‖x‖² ≥ 0 in both conjuncts. In the BInfinite conjunct the subtype norm is the ambient norm, so the same argument applies.", "evidence": [ "ComparatorChallenges/MidpointLenses.lean:328-336: theorem midpoint_lens_main : ⏎ (∀ (H : Finset ForestVertex) (_ : Ancestral ForestAncestor H) ⏎ (x y : X) (_ : x = head H x) (R : ℝ) (_ : 0 ≤ R) ⏎ (_ : ‖x + y‖ ≤ R) (_ : ‖x - y‖ ≤ R), ⏎ ‖tail H y‖ ≤ 2 * Real.sqrt (R ^ 2 - ‖x‖ ^ 2)) ∧ ⏎ (∀ (H : Finset InfiniteVertex) (_ : Ancestral List.IsPrefix H) ⏎ (x y : BInfinite) (_ : x.val = head H x.val) (R : ℝ) (_ : 0 ≤ R) ⏎ (_ : ‖x + y‖ ≤ R) (_ : ‖x - y‖ ≤ R), ⏎ ‖tail H y.val‖ ≤ 2 * Real.sqrt (R ^ 2 - ‖x‖ ^ 2)) :=" ] } ], "readings": [ "first/batch1.json", "flags/flags4.json" ] }, { "family": "333", "title": "Smooth isometric immersions of surfaces into ℝ4", "headline": "Every closed smooth Riemannian surface admits a smooth isometric immersion into ℝ4, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature.", "challenges": [ "SurfaceImmersion" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Every closed smooth Riemannian surface admits a smooth isometric immersion into ℝ4", "coveredBy": [ "SurfaceImmersion" ], "note": "Compact Hausdorff second-countable smooth surface modelled on R² (no boundary) with a smooth metric g: ∃ smooth F into R⁴ with ⟨dF v, dF w⟩ = g(v,w)." }, { "clause": "This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature.", "coveredBy": [ "SurfaceImmersion" ], "note": "No orientability or curvature hypothesis." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "334", "title": "A smooth surface metric with no local isometric immersion in ℝ3", "headline": "Constructs a smooth positive-definite metric on $`(-1,1)^2`$ for which no neighborhood of the origin admits a smooth isometric immersion into ℝ3. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.", "challenges": [ "IsometricImmersion" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a smooth positive-definite metric on $`(-1,1)^2`$ for which no neighborhood of the origin admits a smooth isometric immersion into ℝ3.", "coveredBy": [ "IsometricImmersion" ], "note": "∃ smooth PD symmetric matrix field g on the open square such that, for every open U ⊆ square containing 0, no smooth F : U → ℝ³ has ⟨dF v, dF w⟩ = vᵀ g w everywhere on U." }, { "clause": "This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.", "coveredBy": [ "IsometricImmersion" ], "note": "U ranges over every open neighborhood of 0 inside the square." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "IsometricImmersion", "flag": "JUNK? checked, none: mfderiv would be 0 at non-differentiable points, but F is required to be smooth." } ], "flagReadings": [ { "challenge": "IsometricImmersion", "flag": "JUNK? checked, none: mfderiv would be 0 at non-differentiable points, but F is required to be smooth.", "decision": "cleared", "why": "This flag points in the weakening direction, because the theorem is a negation. A junk-0 mfderiv would break the isometry equation (v ⬝ᵥ g v > 0 for v ≠ 0, since g is PosDef) and make the negated conjunction false for free. But the same conjunction requires ContMDiff … ∞ F (line 51). Smoothness of order ∞ ≥ 1 gives MDifferentiable at every p, so mfderiv is the true derivative wherever the equation is read. Lane K: CERTIFIED.", "evidence": [ "lean/ComparatorChallenges/IsometricImmersion.lean:51: ¬ (ContMDiff 𝓘(ℝ, Coord) 𝓘(ℝ, Ambient) ∞ F ∧", "lean/ComparatorChallenges/IsometricImmersion.lean:53: (inner ℝ : Ambient → Ambient → ℝ) (mfderiv 𝓘(ℝ, Coord) 𝓘(ℝ, Ambient) F p v)" ] } ], "readings": [ "first/batch5.json", "flags/flags1.json" ] }, { "family": "337", "title": "Sharp Cartan–Hadamard isoperimetry and rigidity", "headline": "Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most κ ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for κ = 0 are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral n-cycles, n ≥ 2, in arbitrary proper CAT$`(0)`$ spaces.", "challenges": [ "FillingCoefficient", "SharpCAT0Filling" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most κ ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball.", "status": "unlinked", "coveredBy": [], "manuscript": "Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026", "evidence": [ "corpus/openai-math/release.json:12470: \"dir\": \"Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026\"", "corpus/openai-math/release.json:12472: In a complete simply connected smooth manifold with sectional curvature at most κ ≤ 0, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature κ.", "lean/docs/337.md:3: The following describes the scope of the Lean formalization related to the following accompanying paper(s):", "lean/docs/337.md:5: - [Sharp integral fillings in $\\mathrm{CAT}(0)$ spaces](../../preprints/Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026/paper.pdf)", "lean/docs/337.md:11: This yields the selected integral-current form of the Cartan–Hadamard isoperimetric conjecture." ] }, { "clause": "Bounded positive-volume equality regions for κ = 0 are Euclidean balls.", "status": "unlinked", "coveredBy": [], "manuscript": "Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026", "evidence": [ "corpus/openai-math/release.json:12472: For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null sets with an open region isometric, with its induced metric, to a round Euclidean ball.", "lean/docs/337.md:5: - [Sharp integral fillings in $\\mathrm{CAT}(0)$ spaces](../../preprints/Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026/paper.pdf)" ] }, { "clause": "Also proves sharp Euclidean filling bounds for compactly supported integral n-cycles, n ≥ 2, in arbitrary proper CAT$`(0)`$ spaces.", "status": "stated", "coveredBy": [ "SharpCAT0Filling", "FillingCoefficient" ], "evidence": [ "lean/ComparatorChallenges/SharpCAT0Filling.lean:135: theorem sharp_integral_filling", "lean/ComparatorChallenges/SharpCAT0Filling.lean:136: (X : Type u) [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X]", "lean/ComparatorChallenges/SharpCAT0Filling.lean:137: (hX : IsCAT0 X) (n : ℕ) (hn : 2 ≤ n) (T : IntegralCurrent X n)", "lean/ComparatorChallenges/SharpCAT0Filling.lean:138: (hTc : CompactlySupported T.val) (hTz : IsCycle T.val) :", "lean/ComparatorChallenges/SharpCAT0Filling.lean:139: ∃ S : IntegralCurrent X (n + 1), CompactlySupported S.val ∧", "lean/ComparatorChallenges/SharpCAT0Filling.lean:140: boundarySucc S.val = T.val ∧", "lean/ComparatorChallenges/SharpCAT0Filling.lean:141: mass S.val ≤ fillingCoefficient n * (mass T.val) ^ fillingPower n := by", "lean/ComparatorChallenges/FillingCoefficient.lean:120: theorem coefficient_optimal (n : ℕ) (hn : 2 ≤ n) :", "lean/ComparatorChallenges/FillingCoefficient.lean:121: ∃ T : IntegralCurrent (Euc (n + 1)) n,", "lean/ComparatorChallenges/FillingCoefficient.lean:122: CompactlySupported T.val ∧ IsCycle T.val ∧ 0 < mass T.val ∧", "lean/ComparatorChallenges/FillingCoefficient.lean:123: ∀ S : IntegralCurrent (Euc (n + 1)) (n + 1),", "lean/ComparatorChallenges/FillingCoefficient.lean:124: CompactlySupported S.val → boundarySucc S.val = T.val →", "lean/ComparatorChallenges/FillingCoefficient.lean:125: fillingCoefficient n * (mass T.val) ^ fillingPower n ≤ mass S.val := by", "encoding: Ambrosio–Kirchheim metric currents as functionals with integer-rectifiable charts (SharpCAT0Filling.lean:36-103); CAT(0) via geodesic segments with the comparison inequality (lines 116-124); C_n = 1/((n+1)|S^n|^{1/n}), the Euclidean ball constant (lines 128-133)" ] } ], "gap": "No Riemannian statement: neither the κ ≤ 0 isoperimetric comparison for finite-perimeter sets nor the κ = 0 equality rigidity is stated. (The κ = 0 inequality in dimension ≥ 3 would follow from the filling theorem via Hadamard manifolds being CAT(0) and the currents of finite-perimeter sets, but that bridge is not stated; κ < 0 and rigidity do not follow.)", "scopeNoteOnGap": "silent — the generalized Cartan–Hadamard paper is not linked; the only related sentence is lean/docs/337.md:11: This yields the selected integral-current form of the Cartan–Hadamard isoperimetric conjecture.", "note": "The CAT(0) filling clause is stated, with sharpness. The Riemannian κ ≤ 0 isoperimetry and κ = 0 rigidity are the generalized Cartan–Hadamard manuscript, which the note does not link, hence UNLINKED. Nuance: the linked filling paper also claims the Euclidean (κ = 0, dim ≥ 3) Cartan–Hadamard conjecture, which would follow only through unformalized bridges (Hadamard manifolds are CAT(0), finite-perimeter sets as integral currents, the constancy theorem); 337.md:11 calls what is stated \"the selected integral-current form\", which I read as naming the form, not the omission. I differ from the first reader only through amendment 3.", "firstSuspicious": "Pattern seen across this batch: the CONTENTS.md headline was widened to manuscripts the scope note does not link, and the scope note never says those manuscripts are unformalized. Strictly that is silence, hence UNDECLARED; a second reader should weigh whether the scope note's paper list counts as a declaration. Here: the September 23 generalized Cartan–Hadamard paper; the scope note's 'selected integral-current form' hints at a narrowing without naming the κ < 0 or rigidity claims.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch4.json", "second/second1.json" ] }, { "family": "342", "title": "Donaldson's tamed-to-compatible conjecture", "headline": "Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change.", "challenges": [ "TamingCompatibility" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form.", "coveredBy": [ "TamingCompatibility" ], "note": "Stated, with an added [ConnectedSpace X]; the disconnected case follows componentwise and the scope note says 'every closed connected smooth four-manifold'." }, { "clause": "The almost complex structure stays fixed; the form's cohomology class may change.", "coveredBy": [ "TamingCompatibility" ], "note": "Same J in hypothesis and conclusion; no cohomology constraint on η." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "TamingCompatibility", "flag": "STALE LINK: the scope note links the September-23 edition while the census manuscript is the October-6 edition; the main theorem is unchanged (the October-6 abstract only adds a compatible-cone characterization, not in the headline)" }, { "challenge": "TamingCompatibility", "flag": "SCOPE: [ConnectedSpace X] is a hypothesis not in the headline; harmless (componentwise) and stated in the scope note" } ], "flagReadings": null, "readings": [ "first/batch1.json" ] }, { "family": "343", "title": "Symplectic ball packing in higher dimensions", "headline": "Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension $`2n\\ge6`$. Finitely many closed symplectic balls of capacities $`R_1,\\ldots,R_k`$ embed disjointly into an open ball of capacity R exactly when $`\\sum_iR_i^n\\lt R^n`$ and $`R_i+R_j\\lt R`$ for every distinct pair i, j.", "challenges": [ "BallPacking", "BallPackingNecessity" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension $`2n\\ge6`$. Finitely many closed symplectic balls of capacities $`R_1,\\ldots,R_k`$ embed disjointly into an open ball of capacity R exactly when $`\\sum_iR_i^n\\lt R^n`$ and $`R_i+R_j\\lt R`$ for every distinct pair i, j.", "coveredBy": [ "BallPacking", "BallPackingNecessity" ], "note": "∀ n ≥ 3, k ≥ 1, R > 0, rᵢ > 0: disjoint symplectic embeddings of the closed capacity-rᵢ balls (defined on open neighbourhoods) into the open capacity-R ball exist ↔ Σ rᵢⁿ < Rⁿ and rᵢ + rⱼ < R for i ≠ j." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch3.json" ] }, { "family": "347", "title": "Counterexamples to stable-Morse and strong Arnold fixed-point bounds", "headline": "Disproves stable-Morse lower bounds for nondegenerate Hamiltonian fixed points: on simply connected closed Kähler manifolds of real dimension 22, the deficit below the stable Morse number is unbounded. A separate Hamiltonian diffeomorphism of the complex quadric threefold has exactly three fixed points, fewer than the four critical points required of every smooth function.", "challenges": [ "ArnoldCounterexample" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Disproves stable-Morse lower bounds for nondegenerate Hamiltonian fixed points: on simply connected closed Kähler manifolds of real dimension 22, the deficit below the stable Morse number is unbounded.", "coveredBy": [], "note": "Not stated: the only challenge concerns the quadric threefold. Nothing about Kähler manifolds of real dimension 22, stable Morse numbers or nondegenerate fixed points appears." }, { "clause": "A separate Hamiltonian diffeomorphism of the complex quadric threefold has exactly three fixed points, fewer than the four critical points required of every smooth function.", "coveredBy": [ "ArnoldCounterexample" ], "note": "∃ Hamiltonian φ on Q³ ⊆ ℂP⁴ with exactly 3 fixed points, at least one degenerate, while every smooth function on Q³ has ≥ 4 critical points." } ], "gap": "The stable-Morse clause (unbounded deficit for nondegenerate Hamiltonian fixed points on simply connected closed Kähler 22-manifolds) is not stated.", "scopeNoteOnGap": "The separate nondegenerate Morse-number construction is not included.", "note": null, "firstSuspicious": "The exclusion sentence (347.md:9) names 'the separate nondegenerate Morse-number construction' — the subject of the September 23 12-manifold paper (Morse number) — while the omitted headline clause is the October 5 stable-Morse, real-dimension-22 result, whose manuscripts the scope note does not link (it links only the quadric paper, 347.md:5). Read as DECLARED because the excluded category (nondegenerate Morse-type counterexample) is the omitted clause; a second reader may judge it UNDECLARED.", "confidence": "medium", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch3.json" ] }, { "family": "348", "title": "Nonnegative-curvature Einstein classification and an L2 topological gap", "headline": "Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S4, Fubini–Study $`\\mathbb{CP}^2`$, or a product of equal round two-spheres. A closed simply connected nonnegatively curved four-manifold is diffeomorphic to one of these whenever the scale-invariant L2 norm of its trace-free Ricci curvature lies below a universal positive constant.", "challenges": [ "EinsteinFour" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S4, Fubini–Study $`\\mathbb{CP}^2`$, or a product of equal round two-spheres.", "status": "unlinked", "coveredBy": [ "EinsteinFour" ], "manuscript": "Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026", "evidence": [ "lean/ComparatorChallenges/EinsteinFour.lean:83: 0 < sectionalNumerator g c (chartAt FourSpace c c) u v", "lean/ComparatorChallenges/EinsteinFour.lean:211: (hEinstein : IsEinstein g) (hpositive : HasPositiveSectionalCurvature g) :", "lean/docs/348.md:5: - [Positively curved Einstein four-manifolds](../../preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf)", "CONTENTS.md:8458: We prove that a closed Einstein four-manifold with positive Einstein constant, nonnegative sectional curvature, and a zero-curvature plane has universal Riemannian cover isometric to a product of two round two-spheres." ], "note": "The strictly-positive subcase is stated, and more strongly (M itself is, up to scaling, S⁴, CP² or RP⁴, so its cover is S⁴ or CP²). The nonnegative case with a zero-curvature plane, the S²×S² alternative, is not stated; it comes from the October-4 zero-plane manuscript, which the note does not link." }, { "clause": "A closed simply connected nonnegatively curved four-manifold is diffeomorphic to one of these whenever the scale-invariant L2 norm of its trace-free Ricci curvature lies below a universal positive constant.", "status": "unlinked", "coveredBy": [], "manuscript": "An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026", "evidence": [ "lean/docs/348.md:5: - [Positively curved Einstein four-manifolds]", "CONTENTS.md:8467: If the trace-free Ricci energy is sufficiently small, the manifold is diffeomorphic to S4, $`\\mathbb{CP}^2`$, or $`S^2\\times S^2`$." ], "note": "No L² norm, trace-free Ricci energy or diffeomorphism conclusion appears." } ], "gap": "The nonnegative-curvature classification (zero-plane rigidity: universal cover S²×S² with equal radii) and the L² trace-free-Ricci gap theorem — the two October manuscripts, neither linked by the scope note.", "scopeNoteOnGap": "silent", "note": "The positive-curvature classification (the linked paper) is stated. Both missing pieces (the zero-plane S²×S² case and the L² gap) come from the October manuscripts, which the note does not link. Flag: META COMMAND IN STATEMENT (two `run_cmd Lean.modifyEnv … auxLemmasExt` lines at EinsteinFour.lean:98 and :133), for lane K. The definitions are not holes. The first reader had the same gap as UNDECLARED, before amendment 3.", "firstSuspicious": "The note links only the September-23 positive-curvature manuscript.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "EinsteinFour", "flag": "NARROWER (undeclared): HasPositiveSectionalCurvature (strict) instead of nonnegative; no L² gap" }, { "challenge": "EinsteinFour", "flag": "STATEMENT-FILE METAPROGRAM: two `run_cmd Lean.modifyEnv … auxLemmasExt.modifyState` lines (aux-lemma cache reset)" } ], "flagReadings": null, "readings": [ "first/batch1.json", "second/second2.json" ] }, { "family": "350", "title": "Yau’s nodal bounds: surfaces and higher dimensions", "headline": "Proves the sharp $`C\\sqrt\\lambda`$ upper bound for nodal length on every fixed smooth closed surface, completing Yau's conjecture there. The upper bound fails for fixed smooth metrics in dimensions three and four, including metrics on S3 arbitrarily close to round. In dimension five, nodal measure can grow faster than $`\\lambda^{1/2+\\varepsilon_0}`$ for some fixed $`\\varepsilon_0\\gt 0`$, ruling out even arbitrarily small power losses.", "challenges": [ "NodalLength", "SmoothYau", "YauCounterexample" ], "word": "NARROWER — UNDECLARED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNDECLARED", "disagreement": null, "clauses": [ { "clause": "Proves the sharp $`C\\sqrt\\lambda`$ upper bound for nodal length on every fixed smooth closed surface", "status": "stated", "coveredBy": [ "NodalLength" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/NodalLength.lean:50: nodalLength u ≤ ENNReal.ofReal (C * Real.sqrt lam)", "lean/ComparatorChallenges/NodalLength.lean:52: theorem Main [CompactSpace M] [ConnectedSpace M]" ] }, { "clause": "completing Yau's conjecture there", "status": "declared", "coveredBy": [], "manuscript": "Sharp-nodal-length-on-smooth-surfaces-September-23-2026", "evidence": [ "lean/docs/350.md:11: The known lower bound is not part of this selected statement." ] }, { "clause": "The upper bound fails for fixed smooth metrics in dimensions three and four, including metrics on S3 arbitrarily close to round.", "status": "stated", "coveredBy": [ "SmoothYau" ], "manuscript": null, "evidence": [ "lean/ComparatorChallenges/SmoothYau.lean:135: IsSmoothNeighborhood gRound N →", "lean/ComparatorChallenges/SmoothYau.lean:136: ∃ g ∈ N, HasUnboundedNodalRatio g 3 := by", "lean/ComparatorChallenges/SmoothYau.lean:141: HasUnboundedNodalRatio g 4 := by" ] }, { "clause": "In dimension five, nodal measure can grow faster than $`\\lambda^{1/2+\\varepsilon_0}`$ for some fixed $`\\varepsilon_0\\gt 0`$, ruling out even arbitrarily small power losses.", "status": "undeclared", "coveredBy": [], "manuscript": "Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026", "evidence": [ "lean/ComparatorChallenges/YauCounterexample.lean:62: Tendsto (fun k ↦ nodalMeasure g (u k) / ENNReal.ofReal (Real.sqrt (lam k)))", "preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/main.tex:77: $\\lambda^{1/2+\\epsilon_0}$ for one fixed $\\epsilon_0>0$.", "preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/sections/introduction.tex:36: {\\lambda_j^{1/2+\\epsilon_0}}\\longrightarrow\\infty.", "lean/docs/350.md:15: while the intrinsic four-dimensional nodal measures divided by the square roots of the eigenvalues tend to infinity." ] } ], "gap": "The dimension-five power-law violation (nodal measure ≥ λ^{1/2+ε₀} along a sequence, for a fixed ε₀ > 0) is not stated; only nodal/√λ → ∞ is.", "scopeNoteOnGap": "silent on the rate. lean/docs/350.md:15 describes exactly the weaker statement: \"... while the intrinsic four-dimensional nodal measures divided by the square roots of the eigenvalues tend to infinity.\" It never says the λ^{1/2+ε₀} rate (the title of the linked 'Power-law violations' paper, line 7) is left out.", "note": "Surface upper bound stated (compact connected smooth surface, metric = Riemannian distance via IsRiemannianManifold; connectedness is needed and is in the abstract). The lower bound behind 'sharp/completing Yau' is declared (350.md:11). Dimensions 3 and 4 are stated. IsRound is a satisfiable hypothesis, so sphere_three is not vacuous. Dimension five states only nodal/√λ → ∞, not the λ^{1/2+ε₀} rate or the refutation of every power loss, and the note describes the weaker statement without naming the rate as omitted: undeclared. The first reader's JUNK question is answered: YauCounterexample has no finiteness guard, so a nodal measure of ⊤ would make the ratio ⊤, but Hardt–Simon (1989) gives finite H⁴ nodal measure for every nonzero eigenfunction of a smooth metric on a compact manifold. The route cannot be used, and the statement is not trivialized. Agrees with the first reader.", "firstSuspicious": "The YauCounterexample statement omits the finiteness guard that SmoothYau carries; see the JUNK flag.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "NodalLength", "flag": "JUNK? checked, none: the chart derivatives are genuine for smooth u and a smooth metric (IsContMDiffRiemannianBundle ∞)." }, { "challenge": "SmoothYau", "flag": "JUNK? checked, none: finiteness of the nodal measure is asserted (≠ ∞), so .toReal is genuine." }, { "challenge": "YauCounterexample", "flag": "JUNK? (cleared mathematically, not by the statement): the ratio is an ℝ≥0∞ division, so nodalMeasure = ⊤ would make it ⊤ trivially. Unlike SmoothYau.lean:108, this statement does not assert finiteness. The trivial route is excluded only because nonzero eigenfunctions on the connected compact S⁴×S¹ have nodal sets of finite H⁴-measure (unique continuation plus finiteness of nodal measure); a second reader should confirm that this classical fact holds for every smooth metric." } ], "flagReadings": null, "readings": [ "first/batch5.json", "second/second3.json" ] }, { "family": "353", "title": "Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample", "headline": "Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid. A smooth entire nonquadratic example in dimension ten makes this range sharp. In dimensions three through nine, the paraboloid classification also holds for connected open locally uniformly convex affine-maximal hypersurfaces complete for the affine Berwald–Blaschke metric.", "challenges": [ "AffineBernstein" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid.", "status": "stated", "coveredBy": [ "AffineBernstein" ], "evidence": [ "lean/ComparatorChallenges/AffineBernstein.lean:76: (hΩconvex : Convex ℝ Ω) (u : Space n → ℝ)", "preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/build/main.tex:68: Let \\(\\Omega\\subset\\R^n\\) be a nonempty open convex domain, and let" ], "encoding": [ "the graph is over a nonempty open convex Ω: the manuscript's standing assumption (main.tex:68), which the headline and abstract leave implicit; for Euclidean-complete locally strictly convex graphs it is automatic by the Hadamard–Sacksteder theorem, which is not in the statement", "locally uniformly convex = positive-definite Hessian at every point", "Euclidean completeness = sequential completeness of the intrinsic C¹-path distance of g = δ + Du⊗Du", "elliptic paraboloid = Ω = ℝⁿ, u = ½xᵀAx + b·x + c with A positive definite, and an affine image of the standard paraboloid" ] }, { "clause": "A smooth entire nonquadratic example in dimension ten makes this range sharp.", "status": "unlinked", "coveredBy": [], "manuscript": "Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026", "evidence": [ "CONTENTS.md:8592: We construct a smooth nonquadratic entire graph in dimension ten that solves the classical affine maximal equation and has positive-definite Hessian everywhere.", "lean/docs/353.md:5: - [The affine Bernstein theorem in dimensions three through nine](../../preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/main.pdf)", "preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/build/main.tex:150: It is not smooth and does not furnish a counterexample under the", "lean/docs/353.md:1: # Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample" ], "why": "The linked paper does not claim a smooth dimension-ten example (it says Trudinger–Wang's is not smooth); the claim is only the unlinked October 5 paper's. The note's title names it, which neither declares nor links." }, { "clause": "In dimensions three through nine, the paraboloid classification also holds for connected open locally uniformly convex affine-maximal hypersurfaces complete for the affine Berwald–Blaschke metric.", "status": "declared", "coveredBy": [], "manuscript": "The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026", "evidence": [ "lean/docs/353.md:11: The paper's extension to immersed hypersurfaces without an initial graph assumption is outside this selected statement.", "preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/build/main.tex:51: The same conclusion holds, without an initial graph assumption, for connected" ] } ], "gap": "Not stated: the dimension-ten counterexample (clause 2, undeclared) and the affine-complete hypersurface extension (clause 3, declared).", "scopeNoteOnGap": "Clause 3: docs/353.md:11: \"The paper's extension to immersed hypersurfaces without an initial graph assumption is outside this selected statement.\" Clause 2: silent. The note links only the 3-9 paper (line 5); the dimension-ten counterexample appears only in the note's title (line 1), and no sentence says it is not formalized.", "note": "The Euclidean-complete graph theorem is stated as the linked paper states it (convex domain named as its encoding). The affine-complete hypersurface extension is declared by the note, and the dimension-ten counterexample comes only from the unlinked October 5 paper. So NARROWER — UNLINKED. I read the clauses as the first reader did; the word differs only because UNLINKED did not exist for them.", "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "AffineBernstein", "flag": "NARROWER — UNDECLARED: the dimension-ten counterexample is a headline clause absent from the Lean (no declaration other than lines 73-87) and absent from the scope note body (docs/353.md:5, 9, 11)." } ], "flagReadings": null, "readings": [ "first/batch7.json", "second/second4.json" ] }, { "family": "354", "title": "The isoperimetric profile of the cubic three-torus", "headline": "Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are $`4\\pi/81`$ and $`1/\\pi`$, with exactly the adjacent two types minimizing at each transition.", "challenges": [ "CubicTorus" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements.", "coveredBy": [ "CubicTorus" ], "note": "For every V ∈ (0,1): a minimizer exists, every minimizer has perimeter = candidateProfile V, and E is a minimizer iff it is (up to a null set and a torus isometry) the canonical ball/tube/slab, or the complement of one at volume 1 − V." }, { "clause": "The transition volumes are $`4\\pi/81`$ and $`1/\\pi`$, with exactly the adjacent two types minimizing at each transition.", "coveredBy": [ "CubicTorus" ], "note": "SmallStandardRegion uses closed ranges V ≤ 4π/81, 4π/81 ≤ V ≤ 1/π, 1/π ≤ V, so both adjacent types are included at each tie, and the ↔ excludes any third type. I checked the transitions from candidateProfile: (36π)^{1/3}v^{2/3} = 2√(πv) at v = 4π/81, and 2√(πv) = 2 at v = 1/π." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch6.json" ] }, { "family": "356", "title": "Gigli’s characterization of Alexandrov curvature", "headline": "Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support $`\\mathop{\\mathrm{RCD}}\\nolimits ((n-1)\\kappa,n)`$ condition with reference measure $`\\mathcal H^n`$ and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced.", "challenges": [ "WeakHessian" ], "word": "DIFFERENT", "firstWord": "DIFFERENT", "secondWord": "DIFFERENT", "disagreement": null, "clauses": [ { "clause": "Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support $`\\mathop{\\mathrm{RCD}}\\nolimits ((n-1)\\kappa,n)`$ condition with reference measure $`\\mathcal H^n`$ and distributional sectional curvature at least κ in the original global test classes.", "status": "unlinked", "coveredBy": [], "manuscript": "Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026", "evidence": [ "lean/ComparatorChallenges/WeakHessian.lean:220: theorem every_geodesic [CompleteSpace X] [TopologicalSpace.SeparableSpace X]", "lean/ComparatorChallenges/WeakHessian.lean:226: (σ : Curve X) (hσ : IsGeodesic σ) : CurveSecondDerivativeBound F G σ := by", "CONTENTS.md:8649: we prove that a complete separable metric space is an n-dimensional Alexandrov space of curvature at least κ if and only if, with reference measure $`\\mathcal H^n`$, it is a full-support $`\\mathrm{RCD}((n-1)\\kappa,n)`$ space whose distributional sectional curvature is at least κ in Gigli's original global test classes.", "lean/docs/356.md:1: # Gigli’s characterization of Alexandrov curvature", "lean/docs/356.md:5: - [Weak Hessian bounds along every geodesic in RCD spaces](../../preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026/weak-hessian-geodesics.pdf)", "lean/docs/356.md:9: The formalization transfers a weak Hessian upper bound to every prescribed minimizing geodesic in an $\\mathrm{RCD}(K,N)$ space with finite $N>1$." ] }, { "clause": "The RCD condition is unreduced.", "status": "unlinked", "coveredBy": [], "manuscript": "Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026", "evidence": [ "lean/ComparatorChallenges/WeakHessian.lean:200: def RCD (m : Measure X) (K N : ℝ) : Prop :=", "CONTENTS.md:8640: The RCD condition is unreduced.", "lean/docs/356.md:5: - [Weak Hessian bounds along every geodesic in RCD spaces](../../preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026/weak-hessian-geodesics.pdf)" ] } ], "gap": "The challenge states a supporting result (on a full-support RCD(K,N) space with 1 < N < ∞, a bounded Lipschitz F with weak Hessian ≤ G·g has (F∘γ)'' ≤ G(γ)·d(γ₀,γ₁)² distributionally along every constant-speed geodesic). It is an ingredient of the characterization, but neither implies it nor is implied by it. Gigli's characterization (the headline) is not stated.", "scopeNoteOnGap": "silent. The note's title is \"Gigli’s characterization of Alexandrov curvature\" (lean/docs/356.md:1), but it links only the weak-Hessian paper (line 5) and describes only that result (line 9); it does not say the characterization is left out.", "note": "The only challenge states the weak-Hessian lemma of the companion paper (a weak Hessian bound transfers to every geodesic in an RCD(K,N) space, 1 < N < ∞); it neither implies nor is implied by Gigli's characterization, and no Alexandrov space, H^n or distributional sectional curvature appears — DIFFERENT. Both headline clauses come from the Gigli-characterization manuscript, which docs/356.md does not link (it links only the weak-Hessian paper), so the release's scope note never claimed the headline; 'unreduced' qualifies the characterization's RCD condition (WeakHessian's own RCD is the unreduced τ-coefficient one, but no characterization uses it). Agrees with the first reader.", "firstSuspicious": "The catalogue marks WeakHessian as a main result (mainResultInCatalogue: true), but the family headline is Gigli's characterization, which no challenge states.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "WeakHessian", "flag": "LARGE-ish (234 lines): read through the closure of every_geodesic: IsGeodesic, metricSpeed, IsAC2, IsTestPlan, BoundedTracks, IsLocalWeakUpperGradient/IsMinimal…, weakGradient, LocalSobolev/Sobolev, gamma, IsLaplacian/laplacian, IsTestFunction, weakHessian, WeakHessianUpperBound, localSlope, cheegerEnergy, QuadraticCheegerEnergy, sigmaCoeff/tauCoeff, CurvatureDimension (couplings, interpolation, renyiIntegral), RCD, FullSupport, FiniteOnBoundedSets, CurveSecondDerivativeBound." }, { "challenge": "WeakHessian", "flag": "VACUITY? checked, none found: RCD as defined is satisfiable (e.g. Euclidean space with Lebesgue measure is RCD(0, n))." } ], "flagReadings": null, "readings": [ "first/batch5.json", "second/second5.json" ] }, { "family": "358", "title": "A three-manifold without conjugate points or nonpositive curvature", "headline": "Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature. This answers negatively, already in dimension three, whether the first metric-existence property implies the second.", "challenges": [ "ConjugatePoints" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature.", "coveredBy": [ "ConjugatePoints" ], "note": "compact connected Hausdorff second-countable smooth 3-manifold with an oriented atlas, a smooth Riemannian metric with no conjugate points (all geodesics on all open intervals, all Jacobi fields), and no smooth metric with g(R(u,v)v,u) <= 0 in every chart" }, { "clause": "This answers negatively, already in dimension three, whether the first metric-existence property implies the second.", "coveredBy": [ "ConjugatePoints" ], "note": "restatement" } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "ConjugatePoints", "flag": "JUNK? cleared (direction checked): fderiv of a non-differentiable function and Matrix.inv of a singular matrix are 0 in Lean; such junk would push curvature toward 0, i.e. make NonpositiveSectionalCurvature EASIER to satisfy, so it cannot make the negated conjunct trivially true" }, { "challenge": "ConjugatePoints", "flag": "VACUITY? cleared: IsGeodesicOn and IsJacobiFieldOn are satisfiable (e.g. J = γ'), so NoConjugatePoints is not vacuously true for every metric" } ], "flagReadings": [ { "challenge": "ConjugatePoints", "flag": "JUNK? cleared (direction checked): fderiv of a non-differentiable function and Matrix.inv of a singular matrix are 0 in Lean; such junk would push curvature toward 0, i.e. make NonpositiveSectionalCurvature EASIER to satisfy, so it cannot make the negated conjunct trivially true", "decision": "cleared", "why": "No junk value is triggered where NonpositiveSectionalCurvature is evaluated, whatever its direction would be. Those points are x ∈ c.target with c ∈ atlas, and c.target is open. mfderiv I3 I3 c.symm x is genuine because c.symm is smooth on c.target. metricMatrix g c y is the coordinate matrix of g in chart c: the preferred-chart identifications in TangentSpace and mfderiv cancel. It is smooth in y on c.target and positive definite, so ⁻¹ is the true inverse. christoffel and curvatureCoefficient take fderiv of functions smooth on the open c.target at points of c.target, so every coordPartial is a genuine partial derivative. I also checked the formulas. christoffel g c x i j k = Γ^i_{jk}, and curvatureCoefficient l k i j = R^l_{kij} for R(X,Y)Z = ∇X∇YZ − ∇Y∇XZ − ∇[X,Y]Z. The predicate sums g(R(u,v)v, u) ≤ 0, which is the correct sign for sectional curvature ≤ 0 (positive on round spheres). So NonpositiveSectionalCurvature is exactly nonpositive sectional curvature, and ¬∃ g is not weakened. The first reader's direction argument is not needed. A partial junk value could in principle have either sign, but none occurs.", "evidence": [ "lean/ComparatorChallenges/ConjugatePoints.lean:29: def metricMatrix (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3) (x : E3) :", "lean/ComparatorChallenges/ConjugatePoints.lean:31: fun i j => g.inner (c.symm x)", "lean/ComparatorChallenges/ConjugatePoints.lean:32: (mfderiv I3 I3 c.symm x (basisVector i))", "lean/ComparatorChallenges/ConjugatePoints.lean:36: fderiv ℝ f x (basisVector i)", "lean/ComparatorChallenges/ConjugatePoints.lean:40: (1 / 2 : ℝ) * ∑ l : Fin 3, (metricMatrix g c x)⁻¹ i l *", "lean/ComparatorChallenges/ConjugatePoints.lean:51: coordPartial (fun y => christoffel g c y l j k) x i -", "lean/ComparatorChallenges/ConjugatePoints.lean:53: ∑ a : Fin 3, (christoffel g c x l i a * christoffel g c x a j k -", "lean/ComparatorChallenges/ConjugatePoints.lean:62: ∀ (c : OpenPartialHomeomorph M E3), c ∈ atlas E3 M →", "lean/ComparatorChallenges/ConjugatePoints.lean:63: ∀ x ∈ c.target, ∀ u v : E3,", "lean/ComparatorChallenges/ConjugatePoints.lean:65: metricMatrix g c x i j * curvatureTerm g c x u v v i * u j) ≤ 0" ] }, { "challenge": "ConjugatePoints", "flag": "VACUITY? cleared: IsGeodesicOn and IsJacobiFieldOn are satisfiable (e.g. J = γ'), so NoConjugatePoints is not vacuously true for every metric", "decision": "cleared", "why": "IsGeodesicOn and IsJacobiFieldOn are the genuine equations. For s ∈ U (open) with γ s ∈ c.source (open), c ∘ γ and the coordinate field are smooth near s (γ is continuous, and fieldCoordinates is ContDiffOn on the open U ∩ γ⁻¹ c.source). The deriv terms and the deriv of alongDerivative at s are therefore genuine. The equations read (c∘γ)'' + Γ((c∘γ)', (c∘γ)') = 0 and D_sD_sJ + R(J, γ')γ' = 0, which is the Jacobi equation for this sign of R. NoConjugatePoints is therefore exactly 'no nontrivial Jacobi field along a geodesic vanishes at two parameters', on every open interval and including constant geodesics (docs/358.md:9). Its premises are satisfiable, e.g. J = γ' or J = (s − a)·γ'. It is not vacuously true for every metric: it fails for any metric containing a region isometric to more than a hemisphere of a round 3-sphere, and every closed 3-manifold carries one because M # S³ ≅ M. So ∃ g, NoConjugatePoints g is a genuine claim.", "evidence": [ "lean/ComparatorChallenges/ConjugatePoints.lean:69: deriv w s + connectionTerm g c (x s) (deriv x s) (w s)", "lean/ComparatorChallenges/ConjugatePoints.lean:75: alongDerivative g c (c ∘ γ) (deriv (c ∘ γ)) s = 0", "lean/ComparatorChallenges/ConjugatePoints.lean:84: ContDiffOn ℝ ∞ (fieldCoordinates γ J c) (U ∩ γ ⁻¹' c.source) ∧", "lean/ComparatorChallenges/ConjugatePoints.lean:86: alongDerivative g c (c ∘ γ)", "lean/ComparatorChallenges/ConjugatePoints.lean:87: (alongDerivative g c (c ∘ γ) (fieldCoordinates γ J c)) s +", "lean/ComparatorChallenges/ConjugatePoints.lean:88: curvatureTerm g c (c (γ s)) (fieldCoordinates γ J c s)", "lean/ComparatorChallenges/ConjugatePoints.lean:89: (deriv (c ∘ γ) s) (deriv (c ∘ γ) s) = 0", "lean/ComparatorChallenges/ConjugatePoints.lean:95: ∀ a ∈ U, ∀ b ∈ U, a ≠ b → J a = 0 → J b = 0 → ∀ s ∈ U, J s = 0", "lean/docs/358.md:9: The formalized result separates absence of conjugate points from nonpositive sectional curvature. It constructs a closed connected orientable smooth three-manifold with a smooth Riemannian metric having no conjugate points, while the same manifold admits no smooth metric of everywhere nonpositive sectional curvature. The no-conjugate-points assertion ranges over all geodesics and tangent-bundle Jacobi fields, including constant geodesics and unbounded time intervals. The separate no-focal-points and CAT(0) consequences are not included." ] } ], "readings": [ "first/batch8.json", "flags/flags2.json" ] }, { "family": "359", "title": "Negative Kähler curvature without bounded holomorphic coordinates", "headline": "Constructs a contractible domain in ℂ3 with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below.", "challenges": [ "PinchedKahler" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs a contractible domain in ℂ3 with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting.", "coveredBy": [ "PinchedKahler" ], "note": "MainTheorem: ∃ open nonempty contractible M ⊆ ℂ²×ℂ, a smooth Hermitian PD closed (Kähler) metric g on M, geodesically complete (every geodesic exists for all time inside M), with real sectional curvature in [−B, −A] (0 < A ≤ B); no bounded holomorphic F : M → ℂ³ with nowhere-zero Jacobian; and M is not biholomorphic to a bounded domain." }, { "clause": "A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below.", "coveredBy": [], "note": "Not stated; declared separate." } ], "gap": "The higher-dimensional one-sided example (the companion 'Liouville' paper) is not stated.", "scopeNoteOnGap": "lean/docs/359.md:9: \"This is the negatively pinched construction; the different companion with only one-sided curvature control is separate.\"", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "PinchedKahler", "flag": "DEFINITION check (curvature sign), cleared: I evaluated the 'sectional' formula on the ball metric −log(1−|z|²) at the origin and got −2 on complex lines and −1/2 on totally real planes. The ratio 4 and the negative sign match complex hyperbolic space. The normalization differs from the Riemannian metric Re h by a positive constant, which is immaterial because A and B are existential." }, { "challenge": "PinchedKahler", "flag": "JUNK? checked, none: the denominator 2·realArea is positive for R-linearly independent u, v (the Gram determinant of the PD real metric)." } ], "flagReadings": [ { "challenge": "PinchedKahler", "flag": "JUNK? checked, none: the denominator 2·realArea is positive for R-linearly independent u, v (the Gram determinant of the PD real metric).", "decision": "cleared", "why": "realArea(g p) u v = Re h(u,u)·Re h(v,v) − (Re h(u,v))² is the Gram determinant of the real inner product Re h on ℂ³ ≅ ℝ⁶. Re h is symmetric because g p is Hermitian (IsKahlerMetric conjunct 2) and positive definite (conjunct 3). So realArea > 0 exactly when u, v are ℝ-linearly independent, which IsNegativelyPinched assumes, and IsKahlerMetric is asserted on the same M in both MainTheorem and MetricTransfer. No division by zero is reachable. Even if it were, a junk 0 would give sectional = 0, which violates ≤ −A < 0, so it could only falsify. The Christoffel symbols use (g p)⁻¹, which is the true inverse for the same reason (PD at every point of M, and geodesics stay in M). Lane K: not run yet.", "evidence": [ "ComparatorChallenges/PinchedKahler.lean:44: def hermitianValue (h : CMatrix) (u v : Ambient) : ℂ :=", "ComparatorChallenges/PinchedKahler.lean:45: ∑ i, ∑ j, h i j * coordinates u i * star (coordinates v j)", "ComparatorChallenges/PinchedKahler.lean:46: ", "ComparatorChallenges/PinchedKahler.lean:47: def realArea (h : CMatrix) (u v : Ambient) : ℝ :=", "ComparatorChallenges/PinchedKahler.lean:48: (hermitianValue h u u).re * (hermitianValue h v v).re - (hermitianValue h u v).re ^ 2", "ComparatorChallenges/PinchedKahler.lean:58: def sectional (g : Ambient → CMatrix) (p u v : Ambient) : ℝ :=", "ComparatorChallenges/PinchedKahler.lean:59: ((tensorValue (curvature g p) u u v v).re - (tensorValue (curvature g p) u v u v).re) /", "ComparatorChallenges/PinchedKahler.lean:60: (2 * realArea (g p) u v)", "ComparatorChallenges/PinchedKahler.lean:62: def IsKahlerMetric (M : Set Ambient) (g : Ambient → CMatrix) : Prop :=", "ComparatorChallenges/PinchedKahler.lean:63: ContDiffOn ℝ ∞ g M ∧", "ComparatorChallenges/PinchedKahler.lean:64: (∀ p ∈ M, ∀ i j, g p i j = star (g p j i)) ∧", "ComparatorChallenges/PinchedKahler.lean:65: (∀ p ∈ M, ∀ u : Ambient, u ≠ 0 → 0 < (hermitianValue (g p) u u).re) ∧", "ComparatorChallenges/PinchedKahler.lean:66: (∀ p ∈ M, ∀ i j k, dz (fun q ↦ g q k j) p i = dz (fun q ↦ g q i j) p k)", "ComparatorChallenges/PinchedKahler.lean:78: def IsNegativelyPinched (M : Set Ambient) (g : Ambient → CMatrix) (A B : ℝ) : Prop :=", "ComparatorChallenges/PinchedKahler.lean:79: 0 < A ∧ A ≤ B ∧ ∀ p ∈ M, ∀ u v : Ambient, LinearIndependent ℝ ![u, v] →", "ComparatorChallenges/PinchedKahler.lean:80: -B ≤ sectional g p u v ∧ sectional g p u v ≤ -A", "ComparatorChallenges/PinchedKahler.lean:92: def MainTheorem : Prop :=", "ComparatorChallenges/PinchedKahler.lean:93: ∃ (M : Set Ambient) (g : Ambient → CMatrix) (A B : ℝ),", "ComparatorChallenges/PinchedKahler.lean:94: IsOpen M ∧ M.Nonempty ∧ ContractibleSpace M ∧ IsKahlerMetric M g ∧", "ComparatorChallenges/PinchedKahler.lean:95: GeodesicallyComplete M g ∧ IsNegativelyPinched M g A B ∧", "ComparatorChallenges/PinchedKahler.lean:96: NoBoundedCoordinates M ∧ ¬ BiholomorphicToBoundedDomain M", "ComparatorChallenges/PinchedKahler.lean:138: def MetricTransfer : Prop :=", "ComparatorChallenges/PinchedKahler.lean:139: ∀ c C : ℝ, 0 < c → c ≤ C → ∃ lam0 : ℝ, 0 < lam0 ∧", "ComparatorChallenges/PinchedKahler.lean:140: ∀ φ : Base → ℝ, ContDiffOn ℝ ∞ φ ball → ChartControl φ c C → ChartJets φ C →", "ComparatorChallenges/PinchedKahler.lean:141: ∀ lam : ℝ, lam0 ≤ lam → ContractibleSpace (hartogs φ) ∧", "ComparatorChallenges/PinchedKahler.lean:142: IsKahlerMetric (hartogs φ) (hartogsMetric φ lam) ∧", "ComparatorChallenges/PinchedKahler.lean:143: GeodesicallyComplete (hartogs φ) (hartogsMetric φ lam) ∧", "ComparatorChallenges/PinchedKahler.lean:144: ∃ A B : ℝ, IsNegativelyPinched (hartogs φ) (hartogsMetric φ lam) A B" ] } ], "readings": [ "first/batch5.json", "flags/flags3.json" ] }, { "family": "360", "title": "Weak MTW curvature gives convexity and regular optimal transport", "headline": "On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting. For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous.", "challenges": [ "WeakMTWGlobalSupport", "BiholderTransport" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting.", "coveredBy": [ "WeakMTWGlobalSupport" ], "note": "n ≥ 2, compact, connected, boundaryless (Euclidean model) smooth Riemannian manifold with weak MTW ⇒ every injectivity domain I(x) ⊆ T_xM is convex." }, { "clause": "For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous.", "coveredBy": [ "BiholderTransport" ], "note": "Stronger than stated: one exponent α and constant C work for every pair of densities between fixed bounds; the optimal map is a homeomorphism, unique a.e." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "WeakMTWGlobalSupport", "flag": "JUNK (low): deriv returns 0 where a function is not differentiable, which would make weak MTW hold vacuously at such points; inside the injectivity domain the cost is smooth near (x, exp_x v), so the derivatives are genuine" } ], "flagReadings": null, "readings": [ "first/batch6.json" ] }, { "family": "361", "title": "Failure of integer-degree harmonic dimension comparison", "headline": "Disproves Yau's proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature. For every sufficiently large integer k, a complete smooth metric on ℝ3 has at least $`(k+2)^2`$ independent harmonic functions of growth at most k, exceeding the Euclidean count $`(k+1)^2`$. The metric may depend on k.", "challenges": [ "HarmonicGrowth" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Disproves Yau's proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature.", "status": "stated", "coveredBy": [ "HarmonicGrowth" ], "encoding": "one global chart on Rⁿ; Christoffel symbols, Ricci (sign: round sphere positive) and Laplace–Beltrami in coordinates; completeness of the intrinsic path distance; euclideanDimension(n,k) = C(n+k−1,k) + C(n+k−2,k−1), the dimension of harmonic polynomials of degree ≤ k; one instance ∃ n even ≥ 8, k ≥ 2", "evidence": [ "lean/ComparatorChallenges/HarmonicGrowth.lean:92-96: def MainClaim : Prop := ⏎ ∃ (n k : ℕ) (g : SmoothMetric n), Even n ∧ 8 ≤ n ∧ 2 ≤ k ∧ ⏎ Complete g ∧ RicciNonnegative g ∧ EuclideanNearOrigin g ∧ PositiveSubunitAVR g ∧ ⏎ ∃ u : Fin (euclideanDimension n k + 1) → (Space n → ℝ), ⏎ LinearIndependent ℝ u ∧ ∀ i, HarmonicGrowth g k (u i)", "lean/ComparatorChallenges/HarmonicGrowth.lean:88-89: def euclideanDimension (n k : ℕ) : ℕ := ⏎ (n + k - 1).choose k + (n + k - 2).choose (k - 1)" ] }, { "clause": "For every sufficiently large integer k, a complete smooth metric on ℝ3 has at least $`(k+2)^2`$ independent harmonic functions of growth at most k, exceeding the Euclidean count $`(k+1)^2`$.", "status": "unlinked", "coveredBy": [], "manuscript": "A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026", "evidence": [ "preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/build/paper.tex:24-28: For every \\(4/9 s.f (t', x, v)) (Ici 0) t + ⏎ (∑ i : Fin 3, velocity v i * spatialPartial (fun y => s.f (t, y, v)) x i) + ⏎ (∑ i : Fin 3, (s.E (t, x) + cross (velocity v) (s.B (t, x))) i * ⏎ spatialPartial (fun v' => s.f (t, x, v')) v i) = 0) ∧", "ComparatorChallenges/VlasovMaxwell.lean:82-84: def CompactOnFiniteHorizons (s : Solution) : Prop := ⏎ ∀ T : ℝ, 0 ≤ T → ∃ K : Set Phase, IsCompact K ∧ ⏎ ∀ t ∈ Icc 0 T, tsupport (fun z => s.f (t, z)) ⊆ K", "ComparatorChallenges/VlasovMaxwell.lean:86-88: def Classical (d : Datum) (s : Solution) : Prop := ⏎ ContDiffOn ℝ 1 s.f particleHalfSpace ∧ ⏎ ContDiffOn ℝ 1 s.E fieldHalfSpace ∧ ContDiffOn ℝ 1 s.B fieldHalfSpace ∧", "ComparatorChallenges/VlasovMaxwell.lean:103-105: theorem global_classical_solution (d : Datum) (hd : Admissible d) : ⏎ ∃ s : Solution, Classical d s ∧ SmoothOnFiniteHorizons s ∧ ⏎ ∀ s' : Solution, Classical d s' → SameNonnegativeTime s s' := by" ] } ], "readings": [ "first/batch5.json", "flags/flags4.json" ] }, { "family": "363", "title": "Nonuniqueness with local conservation for the hard-sphere Boltzmann equation", "headline": "Constructs two distinct global entropy solutions of the three-dimensional periodic hard-sphere Boltzmann equation from the same nonnegative initial density, with bounded velocity support and finite mass, energy and absolute entropy. Both are strongly continuous in L1 and satisfy exact local conservation of mass, momentum and kinetic energy.", "challenges": [ "BoltzmannNonuniqueness" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs two distinct global entropy solutions of the three-dimensional periodic hard-sphere Boltzmann equation from the same nonnegative initial density, with bounded velocity support and finite mass, energy and absolute entropy", "coveredBy": [ "BoltzmannNonuniqueness" ], "note": "There exist f0 with bounded velocity support and two AdmissibleGlobal (renormalized, entropy-class) solutions F and G from f0 that differ a.e. at some t in [0,T]. FiniteEntropyDatum gives finite mass, energy and |f log f|." }, { "clause": "Both are strongly continuous in L1", "coveredBy": [ "BoltzmannNonuniqueness" ], "note": "Only on a common initial interval [0,T] (StrongEarly), not for all times." }, { "clause": "and satisfy exact local conservation of mass, momentum and kinetic energy", "coveredBy": [ "BoltzmannNonuniqueness" ], "note": "Only local mass conservation (local_mass) is stated. Momentum is conserved only in total (momentum_conservation), and energy only as an inequality (energy_inequality)." } ], "gap": "The October refinement in the headline is not stated: exact LOCAL conservation of momentum and kinetic energy, and strong L1 continuity for all times. The Lean states the September result: local mass, total momentum, the energy and entropy inequalities, and strong L1 continuity on [0,T].", "scopeNoteOnGap": "docs/363.md:9: \"Both solutions conserve local mass and total momentum and satisfy the global energy and entropy-dissipation inequalities.\" docs/363.md:11: \"On a common initial interval they are strongly continuous in $L^1$, with integrable collision gains and losses. These are the conditions of the selected periodic result; later local-conservation refinements are outside it.\"", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "BoltzmannNonuniqueness", "flag": "NARROWER (declared): local momentum and energy conservation, and global strong continuity, are absent (Lean lines 182-186, 194-198; docs/363.md:9, 11)." }, { "challenge": "BoltzmannNonuniqueness", "flag": "META COMMAND IN STATEMENT: line 50 runs `run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}`, which mutates the elaborator's auxiliary-lemma cache inside the challenge file. It has no effect on the theorem's meaning, but a statement file that runs environment-modifying meta code deserves a lane-K look." } ], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "365", "title": "Joint metric and connection recovery from one boundary patch", "headline": "Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least three, up to diffeomorphism and gauge fixed on that patch. Inputs and observations use the same patch. In contrast, distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball can have identical full-boundary data.", "challenges": [ "Conductivity" ], "word": "NARROWER — UNLINKED", "firstWord": "NARROWER — UNDECLARED", "secondWord": "NARROWER — UNLINKED", "disagreement": { "first": "NARROWER — UNDECLARED", "second": "NARROWER — UNLINKED", "kind": "the word only: the first readers had no UNLINKED (amendment 3)" }, "clauses": [ { "clause": "Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least three, up to diffeomorphism and gauge fixed on that patch.", "status": "unlinked", "coveredBy": [], "manuscript": "Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026", "evidence": [ "CONTENTS.md:8845: [Determination of a metric and a unitary connection from one boundary patch]", "CONTENTS.md:8847: We prove that zero-frequency boundary measurements on any nonempty open boundary patch determine both a smooth Riemannian metric and a smooth unitary connection on the trivial Hermitian rank-two bundle over a compact connected smooth manifold of dimension at least three with smooth boundary.", "lean/docs/365.md:5: - [Nonuniqueness for Bounded Measurable Scalar Conductivities in Three Dimensions](../../preprints/Nonuniqueness-for-Bounded-Measurable-Scalar-Conductivities-in-Three-Dimensions-September-23-2026/paper.pdf)", "lean/docs/365.md:1: # Joint metric and connection recovery from one boundary patch" ], "note": "The family's lead claim; no metric, connection, manifold or boundary patch occurs in the challenge. The note's title names it, but a title alone does not declare; the note links only the scalar nonuniqueness paper. The metric-only part also comes from the unlinked September-24 smooth anisotropic uniqueness manuscript (CONTENTS.md:8856)." }, { "clause": "Inputs and observations use the same patch.", "status": "unlinked", "coveredBy": [], "manuscript": "Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026", "evidence": [ "CONTENTS.md:8847: Both inputs and observations are restricted to the same patch.", "lean/docs/365.md:5: - [Nonuniqueness for Bounded Measurable Scalar Conductivities in Three Dimensions](../../preprints/Nonuniqueness-for-Bounded-Measurable-Scalar-Conductivities-in-Three-Dimensions-September-23-2026/paper.pdf)" ] }, { "clause": "In contrast, distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball can have identical full-boundary data.", "status": "stated", "coveredBy": [ "Conductivity" ], "evidence": [ "lean/ComparatorChallenges/Conductivity.lean:13: def ball : Set R3 := Metric.ball 0 3", "lean/ComparatorChallenges/Conductivity.lean:71: (∀ᵐ x ∂ballMeasure, c ≤ γ₀ x ∧ γ₀ x ≤ C) ∧", "lean/ComparatorChallenges/Conductivity.lean:74: 0 < ballMeasure {x | γ₀ x ≠ γ₁ x} ∧", "lean/ComparatorChallenges/Conductivity.lean:75: IsDirichletToNeumann γ₀ Λ ∧ IsDirichletToNeumann γ₁ Λ" ], "note": "On B(0,3) ⊂ ℝ³; H¹ as the closure of smooth jets, the trace space as H¹/H¹₀, the weak DN map with unique weak solutions built into IsDirichletToNeumann." } ], "gap": "The headline's main theorem — joint recovery of a smooth metric and unitary connection from one boundary patch (and the metric-only same-patch Calderón uniqueness) — is not formalized; only the bounded-measurable scalar nonuniqueness example is.", "scopeNoteOnGap": "silent", "note": "Only the \"In contrast\" scalar nonuniqueness example is stated; the headline's lead theorem (joint metric and connection recovery from one patch) comes from the October-5 manuscript docs/365.md does not link, so it was never claimed: NARROWER — UNLINKED (not DIFFERENT, because one clause is stated). The first reader found the same gap and called it UNDECLARED before amendment 3; the difference is the amendment. Worth showing on the page: the formalization covers the family's side remark, not its title claim.", "firstSuspicious": "The note links only the September-23 nonuniqueness manuscript; the headline's lead claim is from the October-5 manuscript.", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "Conductivity", "flag": "NARROWER (undeclared): only the counterexample clause" } ], "flagReadings": null, "readings": [ "first/batch1.json", "second/second7.json" ] }, { "family": "367", "title": "The critical dimension for the one-phase Bernoulli problem", "headline": "Establishes seven as the first dimension admitting a nonflat, one-homogeneous global minimizer of the one-phase Bernoulli energy. Consequently, minimizing free boundaries are smooth through dimension six, and their singular sets have dimension at most $`n-7`$ in higher dimensions.", "challenges": [ "BernoulliNonflatInSeven" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Establishes seven as the first dimension admitting a nonflat, one-homogeneous global minimizer of the one-phase Bernoulli energy.", "coveredBy": [ "BernoulliNonflatInSeven" ], "note": "Only the existence half: a nonzero, one-homogeneous, non-flat global minimizer in ℝ⁷. 'First' (every such minimizer in dimensions ≤ 6 is flat) is not stated." }, { "clause": "Consequently, minimizing free boundaries are smooth through dimension six, and their singular sets have dimension at most $`n-7`$ in higher dimensions.", "coveredBy": [], "note": "Not stated." } ], "gap": "Flatness of one-homogeneous global minimizers in dimensions ≤ 6 (the 'first' in the headline) and the regularity and singular-set consequences.", "scopeNoteOnGap": "lean/docs/367.md:11: \"The flatness classification in dimensions at most six and the resulting regularity and singular-set bounds are outside this selected statement.\"", "note": null, "firstSuspicious": "The formalized half — a nonflat minimizing cone in ℝ⁷ — is, to my knowledge, the 2009 De Silva–Jerison example ('A singular energy minimizing free boundary', J. reine angew. Math. 635); the paper's new content (flatness in dimensions 5 and 6) is the declared-outside half. A second reader should confirm the prior-art attribution before it is stated anywhere.", "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch6.json" ] }, { "family": "369", "title": "The hot spots conjecture for simply connected planar domains", "headline": "Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed.", "challenges": [ "HotSpots" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed.", "coveredBy": [ "HotSpots" ], "note": "For every nonempty bounded simply connected planar domain with smooth boundary and every u ≢ 0 in the first positive Neumann eigenspace (weak formulation, smooth up to the boundary): ∇u ≠ 0 in Ω, and inf_{∂Ω} u < u(x) < sup_{∂Ω} u for x ∈ Ω. Multiplicity is allowed since any eigenfunction qualifies." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "HotSpots", "flag": "JUNK? (checked, benign): firstPositiveNeumannValue is an sInf over ℝ (line 36); the Rayleigh set is nonempty and bounded below by 0, so it is the genuine μ₂. sInf/sSup of u on frontier Ω (line 47) are over a nonempty compact image." } ], "flagReadings": [ { "challenge": "HotSpots", "flag": "JUNK? (checked, benign): firstPositiveNeumannValue is an sInf over ℝ (line 36); the Rayleigh set is nonempty and bounded below by 0, so it is the genuine μ₂. sInf/sSup of u on frontier Ω (line 47) are over a nonempty compact image.", "decision": "cleared", "why": "The flag matters because firstPositiveNeumannValue enters the hypothesis hu (InFirstNeumannEigenspace): a junk value could have made the hypothesis unsatisfiable and the theorem vacuous. It is not junk. rayleighValues Ω is nonempty: v(x) = x₀ − (the mean of x₀ over Ω), with constant weak gradient e₀, satisfies HasH1Gradient on the bounded open Ω, has mean 0 and has ∫v² > 0. It is bounded below by 0 (a nonnegative integral over a positive one). So Real.sInf is the genuine infimum μ₂, and all integrands are L² or L¹ on a bounded domain, so no Bochner junk. frontier Ω is nonempty and compact (Ω is bounded, open, nonempty and not the whole plane), and u is continuous on closure Ω (ContDiffOn ∞, line 39). So u '' frontier Ω is nonempty and compact, and sInf/sSup are its min and max. Lane K: CERTIFIED.", "evidence": [ "lean/ComparatorChallenges/HotSpots.lean:33: 0 < (∫ x in Ω, (v x) ^ 2) ∧", "lean/ComparatorChallenges/HotSpots.lean:34: r = (∫ x in Ω, ‖g x‖ ^ 2) / (∫ x in Ω, (v x) ^ 2)}", "lean/ComparatorChallenges/HotSpots.lean:36: def firstPositiveNeumannValue (Ω : Set Plane) : ℝ := sInf (rayleighValues Ω)", "lean/ComparatorChallenges/HotSpots.lean:43: firstPositiveNeumannValue Ω * (∫ x in Ω, u x * v x)", "lean/ComparatorChallenges/HotSpots.lean:47: ∀ x ∈ Ω, sInf (u '' frontier Ω) < u x ∧ u x < sSup (u '' frontier Ω)" ] } ], "readings": [ "first/batch3.json", "flags/flags1.json" ] }, { "family": "370", "title": "The Lane–Emden and Hénon–Lane–Emden conjectures", "headline": "Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, $`p,q\\gt 0`$ and real A, B, the system $`-\\Delta u=|x|^A v^p`$, $`-\\Delta v=|x|^B u^q`$ has no positive entire solution when $`(n+A)/(p+1)+(n+B)/(q+1)\\gt n-2`$, with solutions continuous at the origin and classical elsewhere. No symmetry or growth assumption is needed. Known radial existence gives the exact existence criterion for n ≥ 3 and $`A,B\\gt -2`$.", "challenges": [ "HenonEmden" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension.", "coveredBy": [ "HenonEmden" ], "note": "A = B = 0 is the Lane–Emden case." }, { "clause": "For n ≥ 2, $`p,q\\gt 0`$ and real A, B, the system $`-\\Delta u=|x|^A v^p`$, $`-\\Delta v=|x|^B u^q`$ has no positive entire solution when $`(n+A)/(p+1)+(n+B)/(q+1)\\gt n-2`$, with solutions continuous at the origin and classical elsewhere.", "coveredBy": [ "HenonEmden" ], "note": "Exactly the stated class: continuous everywhere, C² off 0, positive, equations off 0." }, { "clause": "No symmetry or growth assumption is needed.", "coveredBy": [ "HenonEmden" ], "note": "None is imposed." }, { "clause": "Known radial existence gives the exact existence criterion for n ≥ 3 and $`A,B\\gt -2`$.", "coveredBy": [], "note": "Not stated, and not a claim of this work: the headline credits the existence direction to known prior work (Bidaut-Véron–Giacomini). Treated as context, not a gap." } ], "gap": null, "scopeNoteOnGap": null, "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "HenonEmden", "flag": "JUNK? checked, none: the Laplacian at x ≠ 0 is the genuine second derivative because u is C² on the open set {0}ᶜ." } ], "flagReadings": [ { "challenge": "HenonEmden", "flag": "JUNK? checked, none: the Laplacian at x ≠ 0 is the genuine second derivative because u is C² on the open set {0}ᶜ.", "decision": "cleared", "why": "Mathlib's Laplacian on E → F is Σᵢ iteratedFDeriv ℝ 2 u x (bᵢ, bᵢ), using the global iteratedFDeriv. For x ≠ 0, ContDiffOn ℝ 2 u {0}ᶜ on the open set {0}ᶜ gives ContDiffAt ℝ 2 u x, so the global iteratedFDeriv equals the classical second derivative at x. The same holds for v. The rpow terms are also genuine: ‖x‖ > 0 for x ≠ 0, and v x, u x > 0. This flag matters because the theorem is a nonexistence claim. A junk Laplacian (0) would turn equation_u into '0 = positive', which no u satisfies, and ¬∃ would be trivially true. That weakening path is closed.", "evidence": [ "lean/ComparatorChallenges/HenonEmden.lean:12: regular_u : ContDiffOn ℝ 2 u ({0}ᶜ : Set (Space n))", "lean/ComparatorChallenges/HenonEmden.lean:13: regular_v : ContDiffOn ℝ 2 v ({0}ᶜ : Set (Space n))", "lean/ComparatorChallenges/HenonEmden.lean:14: positive_u : ∀ x, 0 < u x", "lean/ComparatorChallenges/HenonEmden.lean:15: positive_v : ∀ x, 0 < v x", "lean/ComparatorChallenges/HenonEmden.lean:16: equation_u : ∀ x, x ≠ 0 →", "lean/ComparatorChallenges/HenonEmden.lean:17: -(Laplacian.laplacian u) x = Real.rpow ‖x‖ A * Real.rpow (v x) p", "lean/ComparatorChallenges/HenonEmden.lean:26: ¬ ∃ u v : Space n → ℝ, IsSolution n p q A B u v := by" ] } ], "readings": [ "first/batch5.json", "flags/flags2.json" ] }, { "family": "371", "title": "Stable blowup for the defocusing Schrödinger equation", "headline": "For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in $`H^k(\\mathbb T^{12})`$, with k > 8, whose solutions of the scalar defocusing nonlinear Schrödinger equation blow up in finite time. Thus finite-time blowup is stable under Sobolev perturbations in this supercritical regime.", "challenges": [ "DefocusingNLS" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in $`H^k(\\mathbb T^{12})`$, with k > 8, whose solutions of the scalar defocusing nonlinear Schrödinger equation blow up in finite time.", "coveredBy": [ "DefocusingNLS" ], "note": "stable_blowup: ∀ p₀ ∃ odd p ≥ p₀ (p ≥ 3) ∃ natural k > 8 ∃ nonempty open U in the weighted-ℓ² model of H^k(T¹²) whose classical solutions blow up in finite time at the self-similar rate. Arbitrarily large p, not every large p (declared)." }, { "clause": "Thus finite-time blowup is stable under Sobolev perturbations in this supercritical regime.", "coveredBy": [ "DefocusingNLS" ], "note": "U is open in H^k." } ], "gap": "Arbitrarily large odd powers (∀ p₀ ∃ p ≥ p₀) instead of every sufficiently large odd power; for each such p one natural Sobolev index k > 8 is selected.", "scopeNoteOnGap": "lean/docs/371.md:9: The selected quantifier gives arbitrarily large odd powers, rather than every sufficiently large odd power.", "note": null, "firstSuspicious": "The equation (nonlinearity and linear flow) sits in definition holes, and the file edits the environment with run_cmd: the word is conditional on lane K.", "confidence": "medium-high (the definition holes carry the equation)", "conditionalOnLaneK": [ { "challenge": "DefocusingNLS", "strict": "BODIES DIFFER" } ], "flags": [ { "challenge": "DefocusingNLS", "flag": "DEFINITION HOLE: DefocusingNLS.json definition_names = OAI.DefocusingNLS.sobolevProduct, OAI.DefocusingNLS.schrodingerFlow, OAI.DefocusingNLS.sobolevOddPower — the nonlinearity |u|^{p−1}u and the linear flow e^{itΔ}, i.e. the equation itself, are holes whose bodies Comparator does not compare; the challenge's sobolevProduct body ends in `by sorry` (line 107)" }, { "challenge": "DefocusingNLS", "flag": "QUANTIFIER (declared): ∀ p₀ ∃ odd p ≥ p₀ instead of ∃ p₀ ∀ odd p ≥ p₀; k existential and natural" }, { "challenge": "DefocusingNLS", "flag": "META: the statement file runs `run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}` 15 times between definitions — environment manipulation inside a statement file; lane K should confirm it cannot change what Comparator compares" } ], "flagReadings": null, "readings": [ "first/batch4.json" ] }, { "family": "372", "title": "Global uniqueness in smooth isotropic elasticity", "headline": "Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ3, provided μ > 0 and $`3\\lambda+2\\mu\\gt 0`$ on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required.", "challenges": [ "ElasticityUniqueness" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ3, provided μ > 0 and $`3\\lambda+2\\mu\\gt 0`$ on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required.", "coveredBy": [ "ElasticityUniqueness" ], "note": "∀ bounded connected smooth Ω ⊆ ℝ³ and smooth-up-to-closure (λᵢ, μᵢ) with μᵢ > 0, 3λᵢ + 2μᵢ > 0 on the closure: equal displacement-to-traction pairings ⇒ λ₁ = λ₂ and μ₁ = μ₂ on Ω. No analyticity, closeness to constants or boundary-determination hypothesis." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "ElasticityUniqueness", "flag": "JUNK? (checked, benign): dirichletSolution falls back to f.out when no weak solution with trace f exists (lines 86-91); under μ > 0, 3λ+2μ > 0 (strongly convex energy, Korn) the Dirichlet problem is uniquely solvable, so the fallback is never used for admissible moduli. DN uses an arbitrary representative g.out, which is harmless because u_f annihilates zero-trace directions." } ], "flagReadings": [ { "challenge": "ElasticityUniqueness", "flag": "JUNK? (checked, benign): dirichletSolution falls back to f.out when no weak solution with trace f exists (lines 86-91); under μ > 0, 3λ+2μ > 0 (strongly convex energy, Korn) the Dirichlet problem is uniquely solvable, so the fallback is never used for admissible moduli. DN uses an arbitrary representative g.out, which is harmless because u_f annihilates zero-trace directions.", "decision": "cleared", "why": "The fallback f.out sits inside the HYPOTHESIS DN₁ = DN₂, where junk could in principle weaken the theorem, so existence matters. It holds for every admissible pair and every f. λ, μ are smooth on an open U ⊇ closure Ω, and closure Ω is compact, so they are bounded on Ω; energy is then a bounded bilinear form on H1 (products of L² components times bounded coefficients are integrable, so no Bochner junk). Pointwise, λ(tr ε)² + 2μ|ε|² = 2μ|dev ε|² + ((3λ+2μ)/3)(tr ε)² ≥ min(2μ, 3λ+2μ)|ε|², which is uniformly positive by Admissible. Korn's first inequality and Poincaré on the zero-trace subspace (the closure of TestJets) give coercivity. Lax–Milgram then gives a unique u with trace u = f and WeakSolution: write u = f.out + w with w zero-trace, and uniqueness follows because a zero-trace w with energy(w,w) = 0 is 0. The `else f.out` branch is therefore never taken for admissible moduli, and DN is the genuine Dirichlet-to-Neumann pairing. The representative g.out is immaterial because u_f annihilates zero-trace directions. Lane K: not run yet.", "evidence": [ "ComparatorChallenges/ElasticityUniqueness.lean:38: def Admissible (Ω : Set X) (lam mu : X → ℝ) : Prop :=", "ComparatorChallenges/ElasticityUniqueness.lean:39: SmoothUpTo Ω lam ∧ SmoothUpTo Ω mu ∧", "ComparatorChallenges/ElasticityUniqueness.lean:40: ∀ x ∈ closure Ω, 0 < mu x ∧ 0 < 3 * lam x + 2 * mu x", "ComparatorChallenges/ElasticityUniqueness.lean:78: def energy (Ω : Set X) (lam mu : X → ℝ) (u v : H1 Ω) : ℝ :=", "ComparatorChallenges/ElasticityUniqueness.lean:79: ∫ x, (lam x * div Ω u x * div Ω v x +", "ComparatorChallenges/ElasticityUniqueness.lean:80: 2 * mu x * ∑ i, ∑ j, strain Ω u x i j * strain Ω v x i j)", "ComparatorChallenges/ElasticityUniqueness.lean:81: ∂(volume.restrict Ω)", "ComparatorChallenges/ElasticityUniqueness.lean:82: ", "ComparatorChallenges/ElasticityUniqueness.lean:83: def WeakSolution (Ω : Set X) (lam mu : X → ℝ) (u : H1 Ω) : Prop :=", "ComparatorChallenges/ElasticityUniqueness.lean:84: ∀ v : H1 Ω, HasZeroTrace Ω v → energy Ω lam mu u v = 0", "ComparatorChallenges/ElasticityUniqueness.lean:86: def dirichletSolution (Ω : Set X) (lam mu : X → ℝ)", "ComparatorChallenges/ElasticityUniqueness.lean:87: (f : BoundaryData Ω) : H1 Ω := by", "ComparatorChallenges/ElasticityUniqueness.lean:88: classical", "ComparatorChallenges/ElasticityUniqueness.lean:89: exact if h : ∃ u : H1 Ω, trace Ω u = f ∧ WeakSolution Ω lam mu u", "ComparatorChallenges/ElasticityUniqueness.lean:90: then Classical.choose h", "ComparatorChallenges/ElasticityUniqueness.lean:91: else f.out", "ComparatorChallenges/ElasticityUniqueness.lean:92: ", "ComparatorChallenges/ElasticityUniqueness.lean:93: def DN (Ω : Set X) (lam mu : X → ℝ) : BoundaryData Ω → BoundaryData Ω → ℝ :=", "ComparatorChallenges/ElasticityUniqueness.lean:94: fun f g => energy Ω lam mu (dirichletSolution Ω lam mu f) g.out", "ComparatorChallenges/ElasticityUniqueness.lean:96: def MainClaim : Prop :=", "ComparatorChallenges/ElasticityUniqueness.lean:97: ∀ (Ω : Set X), Domain Ω →", "ComparatorChallenges/ElasticityUniqueness.lean:98: ∀ (lam₁ mu₁ lam₂ mu₂ : X → ℝ),", "ComparatorChallenges/ElasticityUniqueness.lean:99: Admissible Ω lam₁ mu₁ → Admissible Ω lam₂ mu₂ →", "ComparatorChallenges/ElasticityUniqueness.lean:100: DN Ω lam₁ mu₁ = DN Ω lam₂ mu₂ →", "ComparatorChallenges/ElasticityUniqueness.lean:101: (∀ x ∈ Ω, lam₁ x = lam₂ x) ∧ (∀ x ∈ Ω, mu₁ x = mu₂ x)" ] } ], "readings": [ "first/batch3.json", "flags/flags3.json" ] }, { "family": "373", "title": "Nonattainment of the three-marginal Coulomb Monge problem", "headline": "An optimal three-particle Coulomb configuration need not be a deterministic function of the first particle, even for smooth identical spatial densities. The Monge and Kantorovich infima agree, but the Monge infimum is not attained. The same phenomenon occurs for every inverse-power Riesz exponent in each dimension at least two, with a suitable density in each case.", "challenges": [ "CoulombCounterexample" ], "word": "NARROWER — DECLARED", "firstWord": "NARROWER — DECLARED", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "An optimal three-particle Coulomb configuration need not be a deterministic function of the first particle, even for smooth identical spatial densities.", "coveredBy": [ "CoulombCounterexample" ], "note": "∃ smooth compactly supported ρ on ℝ³ (√ρ smooth, compactly supported) with Kantorovich minimum finite and attained but strictly below the cost of every pair of measure-preserving Borel maps." }, { "clause": "The Monge and Kantorovich infima agree, but the Monge infimum is not attained.", "coveredBy": [ "CoulombCounterexample" ], "note": "mongeValue = kantorovichValue, finite-cost preserving maps approach it, and NoMongeOptimizer (strict inequality for every pair)." }, { "clause": "The same phenomenon occurs for every inverse-power Riesz exponent in each dimension at least two, with a suitable density in each case.", "coveredBy": [], "note": "Not stated: only s = 1 in ℝ³." } ], "gap": "The inverse-power (Riesz s > 0) extensions in every dimension d ≥ 2.", "scopeNoteOnGap": "lean/docs/373.md:11: \"This is the three-dimensional Coulomb result; the paper's inverse-power extensions in every dimension are outside this statement.\"", "note": null, "firstSuspicious": null, "confidence": "high", "conditionalOnLaneK": null, "flags": [], "flagReadings": null, "readings": [ "first/batch6.json" ] }, { "family": "374", "title": "Sharp one-third stability of Brenier maps", "headline": "For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy $`\\|T_\\mu-T_\\nu\\|_{L^2(\\rho)}\\le C W_2(\\mu,\\nu)^{1/3}`$ uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound.", "challenges": [ "Brenier" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy $`\\|T_\\mu-T_\\nu\\|_{L^2(\\rho)}\\le C W_2(\\mu,\\nu)^{1/3}`$ uniformly over targets in a fixed compact set", "coveredBy": [ "Brenier" ], "note": "one_third_stability: for d >= 2, a compact convex source with nonempty interior and a compact nonempty container, there is C >= 0 such that every pair of probability targets supported in the container has unique optimal maps T1, T2 with ||T1 - T2||_{L2(rho)} <= C W2^(1/3)." }, { "clause": "The exponent is sharp, even for three-atom targets", "coveredBy": [ "Brenier" ], "note": "one_third_exponent_is_sharp: for every d >= 2, exponent > 1/3 and C > 0, there are two distinct three-atom targets in the cube [-1,1]^d with W2 > 0 and C W2^exponent < ||T1 - T2||, with the uniform measure on the cube as source." }, { "clause": "disproving Letrouit's conjectured square-root bound", "coveredBy": [ "Brenier" ], "note": "Exponent 1/2 > 1/3 is an instance of the sharpness theorem." } ], "gap": null, "scopeNoteOnGap": "no gap", "note": null, "firstSuspicious": "All 12 definitions in the theorem statements are Comparator definition holes. The MATCHES word holds only for the bodies as written in the challenge; whether the solution kept those bodies is lane K's question.", "confidence": "high", "conditionalOnLaneK": [ { "challenge": "Brenier", "strict": "SAME BODIES" } ], "flags": [ { "challenge": "Brenier", "flag": "DEFINITION HOLE: definition_names lists 12 names (OAI.Problem358.E, uniformMeasure, IsSupported, IsCoupling, quadraticPlanCost, wasserstein2, graphPlan, IsQuadraticOptimalMap, IsUniqueQuadraticOptimalMap, mapL2Dist, cube, HasExactlyThreeAtoms). Comparator checks only their types, so a solution body such as mapL2Dist := fun _ _ _ => 0 would make one_third_stability trivial, and wasserstein2 or the optimal-map predicates could be redefined. The bodies in the challenge read as faithful; lane K's strict run must decide the solution's bodies." } ], "flagReadings": null, "readings": [ "first/batch7.json" ] }, { "family": "376", "title": "Universal computation in forced Navier–Stokes flows", "headline": "Constructs viscous incompressible flows starting from rest on a fixed flat three-dimensional domain that perform universal computation under smooth external forcing. A terminating compiler turns a Turing machine and input into a finite program for the force, so a designated particle reaches a fixed region exactly when the machine halts. The viscosity is fixed, positive and computable.", "challenges": [ "SolenoidalSheetPrograms", "NavierStokesAlternating", "NavierStokesVelocity", "BalancedBoxRouting", "BalancedThreeStack", "ForcedNavierStokesComputation" ], "word": "MATCHES", "firstWord": "MATCHES", "secondWord": null, "disagreement": null, "clauses": [ { "clause": "Constructs viscous incompressible flows starting from rest on a fixed flat three-dimensional domain that perform universal computation under smooth external forcing", "coveredBy": [ "ForcedNavierStokesComputation", "BalancedThreeStack", "NavierStokesAlternating", "NavierStokesVelocity" ], "note": "ForcedNavierStokesComputation (= BalancedThreeStack): for every finite single-tape machine M and input w, a smooth divergence-free U with U(0) = 0 on ℝ³, compactly supported (one compact set per (M,w)), solving Navier–Stokes with zero pressure and force f₀ + νf₁ for EVERY ν > 0, unique in a stated comparison class; after time 1 the velocity is 1-periodic and depends only on M. NavierStokesAlternating gives the same on ℝ³ with forces and velocities decaying faster than any power; NavierStokesVelocity gives velocity-field detection on T³ and ℝ²×T." }, { "clause": "A terminating compiler turns a Turing machine and input into a finite program for the force", "coveredBy": [ "ForcedNavierStokesComputation", "BalancedThreeStack", "NavierStokesAlternating", "NavierStokesVelocity" ], "note": "EffectiveFamily: total Computable E, b, R evaluating every mixed derivative at rational points to 2^{-n}, bounding it, and bounding the support radius, all indexed by Encodable.encode (M.code, w); NavierStokesAlternating states Computable compileForce : Machine × List ℕ → (Nat.Partrec.Code)³ whose programs return on every input; NavierStokesVelocity's Represents requires the explicit evaluator to terminate (Dom) at every named point." }, { "clause": "so a designated particle reaches a fixed region exactly when the machine halts", "coveredBy": [ "ForcedNavierStokesComputation", "BalancedThreeStack", "NavierStokesAlternating" ], "note": "(∃ t ≥ 0, X t (4,0,0) ∈ (−1,2)³) ↔ Halts M w, where X is a material flow of U (unique, since U has bounded derivatives); in NavierStokesAlternating the particle at (−1,0,0) enters {x₀ > 0} iff the machine halts, with trajectories asserted unique." }, { "clause": "The viscosity is fixed, positive and computable", "coveredBy": [ "ForcedNavierStokesComputation", "BalancedThreeStack", "NavierStokesAlternating", "NavierStokesVelocity", "SolenoidalSheetPrograms" ], "note": "Every ν > 0 is allowed; for computable ν the force is effectively computable (EffectiveFieldIn ∅ f), and relative to any name of ν otherwise." } ], "gap": null, "scopeNoteOnGap": "silent", "note": null, "firstSuspicious": "BalancedThreeStack and ForcedNavierStokesComputation are the same statement under the same theorem name (two challenge configs for one result). The scope note links the September-27 edition of the box-transport paper while the release carries the October-6 edition (pre-registered STALE LINK).", "confidence": "high", "conditionalOnLaneK": null, "flags": [ { "challenge": "ForcedNavierStokesComputation", "flag": "STALE LINK: docs/376.md:8 links the September-27-2026 edition of 'Incompressible Box Transport and Finite Computation', the paper this challenge (and BalancedThreeStack, BalancedBoxRouting) formalizes; the release's manuscript is the October-6-2026 edition (both directories exist under preprints/)." }, { "challenge": "NavierStokesVelocity", "flag": "LARGE STATEMENT: 1,275 lines. I read the theorem and its definitional closure (basic operators, NavierStokes, Observation, both ComparisonClass structures, PeriodicSpace, SmoothBump/TranslationGates/CenterPaths, Stacks, History, TapeCodes.word/table, Prelude, MachineHistory, MachineTorus, the Expr/Ball evaluator, Array/ChartRecipe, MachineRecipe.force, Represents). No definition in the closure mentions halting, choice or junk-prone limits; halting enters only through the conclusion's iff." }, { "challenge": "NavierStokesVelocity", "flag": "SORRY IN A DEFINITION (outside the closure): ComparatorChallenges/NavierStokesVelocity.lean:454 TapeCodes.half is defined with `w.liftOn word (by sorry)`. It is referenced only by TapeCodes.tape (line 457), which nothing in main_result's closure uses, and the sorry'd obligation (trailing blank digits do not change Nat.ofDigits) is true; lane K should still note that the challenge module itself contains a sorry outside a theorem body." } ], "flagReadings": null, "readings": [ "first/batch2.json" ] } ] }