cert-machine · the monthly ledger

Decided, September 2026

Every published claim this machine decided in September 2026, read from the register: what held, what did not, and what went wrong, in one closed vocabulary. The page is a dated diff of the records — nothing on it is typed.

The month is closed. A row belongs to the month in which its deciding record first held it (the first commit, by git's pickaxe on the record). 18 claims were decided before September 2026 and are on the claims desk, not here.

tl;dr

The finding. 68 published claims decided in September 2026: 48 CERTIFIED, 8 PARTIAL, 5 REPAIRED, 4 MIXED, 2 NEEDS DATA, 1 REFUTED. 20 of them name something that went wrong — 7 narrower scope, 4 tolerance witness, 2 arithmetic slip, 2 data not public, 1 not from the published data, 1 not the optimum, 1 wrong quantity, 1 depends on reading, 1 checker wider than definition; the other 48 hold as printed. 4 rows are one audit of many items (a dataset's keys, a benchmark's counts); the row counts once and names its most frequent defect.
The mechanism. Each row is derived from the record that decided it; a claim with no record gets no row. CERTIFIED and REFUTED are theorems; PARTIAL, MIXED, REPAIRED and NEEDS DATA are compositions of the three verdicts, named on the claims desk.
Check it. node tools/run-claims-ledger.js && node tools/build-report-monthly.js — the register is certs/claims-ledger.json.

decided this month

48 CERTIFIED, 8 PARTIAL, 5 REPAIRED, 4 MIXED, 2 NEEDS DATA, 1 REFUTED.

held as printed

The row names nothing that went wrong. Most published claims hold, and the ledger says so first.

something went wrong

7 narrower scope, 4 tolerance witness, 2 arithmetic slip, 2 data not public, 1 not from the published data, 1 not the optimum, 1 wrong quantity, 1 depends on reading, 1 checker wider than definition.

decided before

Carried on the claims desk; the register holds 132 rows in all.

§1 · the rows

What was decided, in the order it was decided

recorded	claim	claimant	verdict	what went wrong	where
2026‑09‑03	K(11) >= 593 — the May 2025 record	AlphaEvolve (Novikov et al., DeepMind)	CERTIFIED	—	the record
2026‑09‑03	K(4) >= 24 — exact value, Musin 2003	classical (generated here)	CERTIFIED	—	the record
2026‑09‑03	K(8) = 240 — exact value, Levenshtein / Odlyzko–Sloane 1979	classical (generated here)	CERTIFIED	—	the record
2026‑09‑03	K(11) >= 594 — the solved n=594 rung, score-0 winner (solution #1492)	EinsteinArena agents (Bianchi et al. platform)	CERTIFIED	—	the record
2026‑09‑03	K(11) >= 604 — the paper's headline result, credited by Cohn's reference table (2026-06-22)	EinsteinArena (Bianchi, Kwon, Pappu, Zou — arXiv:2606.10402)	CERTIFIED	—	the record
2026‑09‑03	K(11) >= 592 — the 2022 record the AI ladder started from	M. Ganzhinov (arXiv:2207.08266, Highly symmetric lines)	QUEUED	—	the record
2026‑09‑03	K(11) >= 604 — configuration 1 of three	The Station agents (dualverse-ai)	CERTIFIED	—	the record
2026‑09‑03	K(11) >= 604 — configuration 2 of three	The Station agents (dualverse-ai)	CERTIFIED	—	the record
2026‑09‑03	K(11) >= 604 — configuration 3 of three	The Station agents (dualverse-ai)	CERTIFIED	—	the record
2026‑09‑03	construction device: 604 integer D12 vectors whose oblique shadow is configuration 3 (also a valid 604-point direction set in R^12)	The Station agents (dualverse-ai)	CERTIFIED	—	the record
2026‑09‑03	K(11) >= 582 — the pre-2022 record shell, integer norm-4 maximum (Lean-proved maximal by the Station)	classical (Best 1977 class; bytes from the Station bundle)	CERTIFIED	—	the record
2026‑09‑08	f ≥ 0 as 600 values on [−1/4, 1/4]: C₁ ≤ 1.50529397	AlphaEvolve (arXiv:2506.13131)	CERTIFIED	—	the record
2026‑09‑08	f ≥ 0 as 1319 values on [−1/4, 1/4]: C₁ ≤ (no value printed for this file)	AlphaEvolve V2 (arXiv:2511.02864)	CERTIFIED	—	the record
2026‑09‑08	f ≥ 0 as 30000 values on [−1/4, 1/4]: C₁ ≤ 1.50286286	Together AI agents (the EinsteinArena repository)	CERTIFIED	—	the record
2026‑09‑08	f ≥ 0 as 30000 values on [−1/4, 1/4]: C₁ ≤ 1.50286290	TTT-Discover (arXiv:2601.16175)	CERTIFIED	—	the record
2026‑09‑08	n = 21 circles in a rectangle of perimeter 4 (bounding box w + h ≤ 2), no overlaps: Σ r = 2.3658321334	AlphaEvolve V2 (Georgiev, Gómez-Serrano, Tao, Wagner — arXiv:2511.02864)	CERTIFIED	—	the record
2026‑09‑08	n = 21 circles in a rectangle of perimeter 4 (bounding box w + h ≤ 2), no overlaps: Σ r = 2.3658323759	Together AI agents (the EinsteinArena repository)	REPAIRED	tolerance witness	the record
2026‑09‑08	100 rows of 20 weights → (edge density, triangle density) points; the platform's envelope score = -0.712494	AlphaEvolve V2 (arXiv:2511.02864)	CERTIFIED	—	the record
2026‑09‑08	100 rows of 20 weights → (edge density, triangle density) points; the platform's envelope score = -0.712256	Together AI agents (the EinsteinArena repository)	CERTIFIED	—	the record
2026‑09‑08	70 coefficients ±1: C⁺ = max_{\|z\|=1} \|g(z)\| / √71 = 1.340925	AlphaEvolve V2 (arXiv:2511.02864)	CERTIFIED	—	the record
2026‑09‑08	70 coefficients ±1: C⁺ = max_{\|z\|=1} \|g(z)\| / √71 = 1.280932	Together AI agents (the EinsteinArena repository)	CERTIFIED	—	the record
2026‑09‑08	n = 14 points, min triangle area / convex-hull area = 0.0278355715	AlphaEvolve V2 (arXiv:2511.02864)	CERTIFIED	—	the record
2026‑09‑08	n = 14 points, min triangle area / convex-hull area = 0.0278355805	Together AI agents (the EinsteinArena repository)	CERTIFIED	—	the record
2026‑09‑08	12 unit hexagons inside a hexagon of side 3.9419123, no overlaps, all inside: score = the outer side	AlphaEvolve V2 (arXiv:2511.02864)	CERTIFIED	—	the record
2026‑09‑08	12 unit hexagons inside a hexagon of side 3.9416523, no overlaps, all inside: score = the outer side	Together AI agents (the EinsteinArena repository)	CERTIFIED	—	the record
2026‑09‑08	n = 16 points, (max distance / min distance)² = 12.889266	AlphaEvolve (Novikov et al. — arXiv:2506.13131)	CERTIFIED	—	the record
2026‑09‑08	n = 16 points, (max distance / min distance)² = 12.889230	Together AI agents (the EinsteinArena repository)	CERTIFIED	—	the record
2026‑09‑08	a step function h with 95 steps, 0 ≤ h ≤ 1, Σh = n/2: upper bound 0.380924	AlphaEvolve (arXiv:2506.13131)	REPAIRED	tolerance witness	the record
2026‑09‑08	a step function h with 51 steps, 0 ≤ h ≤ 1, Σh = n/2: upper bound 0.380927	J. K. Haugland (arXiv:1609.08000)	CERTIFIED	—	the record
2026‑09‑08	a step function h with 600 steps, 0 ≤ h ≤ 1, Σh = n/2: upper bound 0.380871	Together AI agents (the EinsteinArena repository)	REPAIRED	tolerance witness	the record
2026‑09‑08	a step function h with 600 steps, 0 ≤ h ≤ 1, Σh = n/2: upper bound 0.380876	TTT-Discover (Yuksekgonul et al. — arXiv:2601.16175)	REPAIRED	tolerance witness	the record
2026‑09‑15	the exceedance counts printed for the environmental-contour benchmark's 150 contours	the benchmark's contributing teams (Haselsteiner et al., Ocean Eng. 2021)	MIXED	not from the published data	the record
2026‑09‑21	the worked solutions of GSM8K's 8,792 test and train keys	GSM8K (Cobbe et al., 2021)	MIXED	arithmetic slip	the record
2026‑09‑22	the time-horizon coefficients and 50% horizons METR prints for 23 models	METR (Time Horizon 1.1)	MIXED	not the optimum	the record
2026‑09‑27	the 15 Hs marginal fits the benchmark teams printed for buoys A, B and C	the environmental-contour benchmark's contributing teams	MIXED	wrong quantity	the record
2026‑09‑28	the annual-maximum Hs fits of 20 printed rows for five South Atlantic lease areas	Bhaskaran et al. (Energies 16, 6935, 2023)	NEEDS DATA	data not public	the record
2026‑09‑29	Borcea-Branden AIM problems	GPT-5 (Pro), GPT-5.6 (Pro), Opus 5 (S. Sra's counterexample library)	PARTIAL	narrower scope	the record
2026‑09‑29	Courtade's volume conjecture for Minkowski sums is false	Suvrit Sra, by hand (S. Sra's counterexample library)	CERTIFIED	—	the record
2026‑09‑29	Feasible Picard steps for DPP likelihood	GPT-5.6 (S. Sra's counterexample library)	PARTIAL	depends on reading	the record
2026‑09‑29	Failure of the proposed Hamiltonian NEPv Rayleigh identity	OpenAI Codex (S. Sra's counterexample library)	CERTIFIED	—	the record
2026‑09‑29	Log-volume midpoint gap and its Lorentzian generalization	GPT-5.6 Pro (S. Sra's counterexample library)	PARTIAL	narrower scope	the record
2026‑09‑29	Macdonald lattice Schur-convexity	GPT-5.6 (Pro) (S. Sra's counterexample library)	CERTIFIED	—	the record
2026‑09‑29	No dimension-free constant in O'Donnell's matrix conjecture	GPT-5.6 Pro (S. Sra's counterexample library)	PARTIAL	narrower scope	the record
2026‑09‑29	An oblivious subspace injection need not give relative-error sketch-and-solve	OpenAI Codex (S. Sra's counterexample library)	CERTIFIED	—	the record
2026‑09‑29	Exact QRCP can miss the orthonormal-row conditioning bound	OpenAI Codex (S. Sra's counterexample library)	CERTIFIED	—	the record
2026‑09‑29	Quantum coupon collection: positivity of an alternating sum of inverses	GPT Pro (S. Sra's counterexample library)	CERTIFIED	—	the record
2026‑09‑29	A mixed-norm Cauchy-Schwarz question of Sah, Sawhney, Stoner and Zhao	GPT-5.6 Sol (Pro), Opus 5 (S. Sra's counterexample library)	CERTIFIED	—	the record
2026‑09‑29	Strict diagonal dominance does not ensure diminishing Nyström error reductions	OpenAI Codex (S. Sra's counterexample library)	CERTIFIED	—	the record
2026‑09‑29	Derivatives of the Jacobi-theta kernel	GPT-5.5 (Pro) (S. Sra's counterexample library)	CERTIFIED	—	the record
2026‑09‑29	Variance-sensitive Matrix Spencer	GPT-5.5 (Pro) (S. Sra's counterexample library)	PARTIAL	narrower scope	the record
2026‑09‑29	R(k,k) <= 3.78233^(k+o(k)): Theorem 1 of Gupta–Ndiaye–Norin–Wei holds with G_AI (one more iteration of Theorem 14)	ChatGPT 5.6 Sol, printed by Gupta, Ndiaye, Norin and Wei as preliminary and unverified (arXiv 2407.19026v2)	CERTIFIED	—	the record
2026‑09‑29	three closed forms that "pass the compliance check and match the high-precision reference"	GPT-5.6 Sol Max (HorizonMath, arXiv 2603.15617v2)	NEEDS DATA	data not public	the record
2026‑09‑29	Area(E) = 0.1091479892... < 0.1148103258186177 (the AlphaEvolve baseline)	GPT-5.4 Pro, reproduced by GPT-5.6 Sol Max (HorizonMath, arXiv 2603.15617v2, Appendix A.2)	CERTIFIED	—	the record
2026‑09‑29	R(k,k) <= 3.6960839126^(k+o(k)), improving 3.7992 (Gupta–Ndiaye–Norin–Wei), by a certificate that satisfies their Theorem 14	GPT-5.4 Pro, reproduced by GPT-5.6 Sol Max (HorizonMath, arXiv 2603.15617v2, Appendix A.3)	REFUTED	checker wider than definition	the record
2026‑09‑29	C3b >= 1.77898884 (the registry's asterisked lower bound)	Mosaic Intelligence (2026)	CERTIFIED	—	the record
2026‑09‑29	C3c >= 1.6747338950414058 (the registry's asterisked lower bound)	Y. Lin (2026)	CERTIFIED	—	the record
2026‑09‑29	C3c >= 1.6747338950208249 (superseded in the registry)	Mosaic Intelligence (2026)	CERTIFIED	—	the record
2026‑09‑29	C42 <= 0.6906538 (the registry's asterisked upper bound)	S. Griego (github.com/sebastian-griego/turan-c42-certificate v1.0.0)	PARTIAL	narrower scope	the record
2026‑09‑29	C42 <= 0.688983 (an open pull request to the registry)	A. Röhrig with Codex (teorth/optimizationproblems PR #184, 9 Sep 2026)	PARTIAL	narrower scope	the record
2026‑09‑29	C71 > 6.521845710923046575 (the registry's asterisked lower bound)	Numaro (numaro.tech, "AI Autoresearch"), Zenodo 10.5281/zenodo.21497769	CERTIFIED	—	the record
2026‑09‑29	C84b <= 1.999281 (c >= 0.000719) (the registry's asterisked upper bound)	I. Althoefer with ChatGPT 5.5 (the note, 28 May 2026), as the registry quotes it	REPAIRED	arithmetic slip	the record
2026‑09‑29	Phi, a cubic map of C^11 with det J = -2 that is not injective (the gist that first published it says ChatGPT generated it)	Castañeda, Honorato, Valenzuela-Henríquez (arXiv 2608.05392); the map first published in a gist that says ChatGPT generated it	CERTIFIED	—	the record
2026‑09‑29	X on R^14 (degree 7): JX has spectrum {-1} at every point and X has three zeros, so the weak Markus–Yamabe conjecture fails in dimension 14	Castañeda, Honorato, Valenzuela-Henríquez (arXiv 2608.05392)	CERTIFIED	—	the record
2026‑09‑29	X-hat on R^18 (degree 3): spectrum {-1} everywhere, three zeros — decided as a symbolic identity where the paper's appendix checks one sample point	Castañeda, Honorato, Valenzuela-Henríquez (arXiv 2608.05392)	CERTIFIED	—	the record
2026‑09‑29	F4: a polynomial map of C^4 with det J = -44/9 that is not injective (a counterexample to the Jacobian conjecture), printed in full	S. Gao, with Claude Fable 5 assisting (arXiv 2608.00222)	CERTIFIED	—	the record
2026‑09‑29	F5: a polynomial map of C^4 with det J = 160/29 that is not injective (a counterexample to the Jacobian conjecture), printed in full	S. Gao, with Claude Fable 5 assisting (arXiv 2608.00222)	CERTIFIED	—	the record
2026‑09‑29	F6: the map the printed construction defines (n = 5), det J = -290, not injective	S. Gao, with Claude Fable 5 assisting (arXiv 2608.00222)	CERTIFIED	—	the record
2026‑09‑29	F7: the map the printed construction defines (n = 5), det J = 119377, not injective	S. Gao, with Claude Fable 5 assisting (arXiv 2608.00222)	PARTIAL	narrower scope	the record
2026‑09‑29	G: a polynomial map of C^3 with det J = 2 that is not injective (a counterexample to the Jacobian conjecture), printed in full	S. Gao, with Claude Fable 5 assisting (arXiv 2608.00222)	CERTIFIED	—	the record

§2 · the vocabulary

What went wrong, in words

tolerance witness (4). a witness only within a numerical tolerance; the printed value is the tolerance's.
not from the published data (1). the printed number is not what the published data give, and the arithmetic is not why.
arithmetic slip (2). a printed arithmetic step that does not hold.
not the optimum (1). printed as an optimum, but not the one the definition names (a stop short of it, or a local one).
wrong quantity (1). the printed number is a correct answer to a different question.
data not public (2). the data behind the claim are not public, so no one outside can decide it.
narrower scope (7). decided only in a narrower scope than printed; the rest is out of reach, not wrong.
depends on reading (1). the claim holds under one of two definitions its own text gives, and not under the other.
checker wider than definition (1). the checker that accepted the claim admits what the definition it encodes excludes.

The vocabulary is closed: 14 kinds, defined once in tools/run-claims-ledger.js. A defect that fits none of them refuses the register until the vocabulary is decided on, never "other".

§3 · not here

What this page does not count

Refusals of our own. What the instruments would not decide is on the refusals page, by kind and without a total.
Model proposals. The eval board grades what models propose to it; a proposal is not a published claim.
Our own results. Records, bounds and censuses this machine produced are not claims someone else published, and are not audited here.