Equilibrium is a geometry — and refusal is how you measure it
Three independent attacks on the same thesis, recorded SIDE BY SIDE so we can watch which positions survive contact with evidence and whether a hybrid rises. The experiment is the FORM: every position carries a decidable test and a status that may change ONLY by evidence, never by preference. A strategy document that cannot be falsified is a mood board.
The left column is what is actually done. The right column is what it means, for someone who does not work in validated numerics. Neither column is the marketing one — if the analogy claims more than the technical step, the analogy is wrong and gets rewritten.
Four badges, no blurring between them, and each names a MECHANISM rather than a status this project awards itself. MUTATION-TESTED — a headless battery in this repo checks it and is mutation-tested: revert the check and it goes red. STANDARD — off-the-shelf or cited, not re-derived here. PROSPECTIVE — pre-registered: the shape of the claim is fixed in advance, the number is not yet computed. OPEN — the honest frontier, known and not done.
Deliberately absent: the words proved and certified as a status. Both are rungs on this project's evidence ladder — rung 5 needs a named human signature, rung 4 needs a ledger certificate record — and no such record covers any row below. What backs the green rows is a mutation-tested battery, which is what MUTATION-TESTED says and all it says. Most of this ledger is OPEN, and that is the honest shape of a programme one day old.
Teal = population 1 gains, oxblood = population 1 gives up; population 2 moves by exactly the opposite. Amber nodes are the exits, which carry no conservation row. Every direction is a cycle: flow goes round and comes back, so both conservation laws and every edge total are untouched. Direction 6 is the two populations swapping between exit 8 and exit 10 — it exists for the same reason the exits carry no row.
| Claim | Status | What backs it |
|---|---|---|
S1's cost is the edge total j1+j2, identical for both populations, with no population index | MUTATION-TESTED | read from the SHIPPING kernel at run time, not transcribed — test-s1-face.py G1–G1c |
| OCCUPANCY: the phenomenon is known at TITLE level — unique totals with non-unique class splits | STANDARD | Boyce & Xie 2013, "Assigning user class link flows uniquely", TR Part A 53:22–35, DOI 10.1016/j.tra.2013.06.002. An entire programme (proportionality, entropy maximisation, PAS) exists to REPAIR this non-uniqueness |
| The dimension count is FOLKLORE — ker of a node–arc incidence matrix IS the circulation space, so k is the cycle rank m − n + c | STANDARD | three independent derivations reach it in under a page: linear algebra (2 lines), graph theory (textbook), and the cycle-swap argument that Bar-Gera's paired-alternative-segment machinery is built from. Not a rediscovery, not unclaimed |
| All six directions are attained at the tested equilibrium — each movable in BOTH signs without a flow going negative | MUTATION-TESTED | test-attainment.py 5/5. Narrowest room [−7.18, +1.59], widest [−42.32, +57.68]; no shared edge at zero, so the point is relatively interior. Raised as a defect in our claim and resolved by TESTING rather than softening — "dimension exactly 6" is earned AT THAT POINT and must always carry the point |
| The S1 equilibrium face has tangent dimension k = 6 | MUTATION-TESTED | exact rational null space. Two mutants fire: treating the exits as ordinary nodes gives k=5, losing exactly the exit-swap direction. A node-count shortcut (11 − 5) agrees on this instance — but see the row below: it is not a formula |
The node-count shortcut shared − conservation nodes is NOT general — it held on the paper instance by luck | MUTATION-TESTED | test-face-general.py — a random layered DAG with 19 shared edges has rank 10, so k = 9 while the shortcut predicts 8. Found by a test written to break our own conjecture. The publishable quantity is the RANK-based dimension; the shortcut must never be stated as a formula |
| The face computation now runs on an arbitrary two-population network | MUTATION-TESTED | face_general.py, exercised on 12 grids and random DAGs; every basis vector satisfies both conservation laws with residual exactly 0 over ℚ. This is the precondition for pointing it at a public benchmark rather than at our own 15 edges |
| Each of the six directions satisfies both Kirchhoff laws, and they are independent | MUTATION-TESTED | residual exactly 0 over ℚ — not a tolerance. G3, G4 |
| This is not an enclosure. Exact arithmetic on an affine system is algebraic correctness, not validated numerics | STANDARD | the literature gate's own terminology contract, applied to us |
| k = 6 holds at a relatively interior point of the face; at a boundary point the local cone is smaller | STANDARD | standard polyhedral geometry; any claim must name which point |
| A monotone VI has a convex solution set, so two distinct equilibria give a whole segment | STANDARD | classical — and the cheapest available upgrade for the nonlinear cases |
| Bordered Krawczyk closes at two points of a face, giving that segment with certificates | PROSPECTIVE | shape fixed in advance: F̃(x) = (F(x), v̂ᵀx − t) at t₁ ≠ t₂. Not computed |
| Verified rank + the constant-rank theorem gives the dimension for a NONLINEAR instance | OPEN | the honest frontier. S1 is affine and needed none of this |
Four independent solves of the SAME problem, projected onto two of the six tangent directions. They are not one point. The totals agree to 4.4e-13 and every edge off the shared subgraph agrees to 4.3e-14 — the whole spread lives inside the predicted face. This is the thesis in one picture: the equilibrium is not a location, it is a set, and the solver lands somewhere on it depending on where it started.
| Claim | Status | What backs it |
|---|---|---|
| Ground truth for S1 exists: the tangent space is known exactly | MUTATION-TESTED | the six rational directions above — the angle test no longer needs an estimate |
| C2 HOLDS FOR S1 — the six cycle directions lie exactly in ker(J) | MUTATION-TESTED | test-c2-s1.py 6/6. S1 equilibrium KKT Jacobian assembled over ℚ (39×39, rank 33); max residual exactly 0 over 6 directions × 39 equations. And PROVABLE, not measured: set (θ¹,θ²) = (d,−d) — the cost rows cancel because the cost sees only the total, both Kirchhoff rows vanish because d is a circulation |
| …and ker(J) is exactly the face — no gauge, nothing to quotient | MUTATION-TESTED | nullity 6, and a constant potential shift is verified NOT in the kernel. I predicted a gauge excess; there is none. Dropping the exit rows — the paper's Remark 6, adopted for CONDITIONING — pins the potentials and removes the gauge before anyone asks. A numerical choice turns out to be what makes the geometric statement clean |
| Scope: this is a claim about geometric refusals only | STANDARD | in mfg-congest the wall is a representation limit with σmin = 0.645 — there is no null direction to find, so C2 there is not unproven but meaningless. The two-kinds-of-refusal split is what makes the claim sayable at all |
| The solver's reseed differences lie in the predicted 6-dimensional tangent span | MUTATION-TESTED | test-tangent-match.py — 4 independent starts, worst movement 18.807, worst projection residual 2.376e-14. Totals 4.37e-13; d₁+d₂ 4.37e-13; non-shared edges 4.26e-14, predicted before any solve. A corrupted span gives 1.88e+01, so the test discriminates |
| …and that agreement is a float agreement, not an enclosure | STANDARD | the algebra is exact over ℚ; the solver's output is double precision. 2.4e-14 is agreement, not an interval statement |
| Once the rank enclosure holds, the tangent space equals ker(Df) | STANDARD | constant-rank theorem. Classical, and it is most of what C2 wanted to claim |
| The bordering direction that restores contraction is operationally the tangent | PROSPECTIVE | shape fixed; the falsifier is a non-kernel direction that also restores contraction |
| The two-sided directional bound — contraction certified < 1 on v̂⊥, identity-like ≥ 1−O(ε) on span(v̂) | OPEN | the publishable core, and the only part not already a corollary. Not attempted |
| angle(refusal direction, true tangent) → 0 across a zoo of known manifolds | OPEN | line · circle · torus · intersecting planes · saddle · cusp · fold. THE falsifier for the whole thesis. Not run |
The wall, seen physically. Each row is one certified-or-attempted solve; colour is the equilibrium density across the torus. At A = 0.3 the crowd is almost flat (max/min = 1.05); by A = 6 it has piled into the potential well (max/min = 2.93) and the depleted band has thinned to min m = 0.5180. The certificate fails where the crowd gets thin — the enclosure needs the density bounded away from zero, and the amber band is where it stops closing. Nothing here is certified: these are float solves, drawn to show WHERE the wall comes from.
A detail worth keeping: at A = 0.3 this computes min m = 0.9737, and the published floor is 0.9736 — rounded DOWN, because a lower bound rounded to nearest is a false floor. The same distinction cost three live pages a correction earlier today.
Six grids. The points fall on a near-straight line against 1/N — first-order convergence, A*(N) ≈ A∞ − 6.4/N. The amber band at the axis is where the limit lies, and its width is the point: two extrapolation models disagree by 0.08, so the wall converges but its value is a bracket, never a number. An earlier reading of three coarse points said this did NOT converge — a 0.2-wide bracket cannot resolve a decaying increment.
This is why extrapolating from inside works. (1−Z₁) falls linearly in A, so four successful certificates predict where it reaches zero — and you never have to run past the wall to find it. It also refutes a prediction made here earlier: because the residual Y₀ is negligible, the radii-polynomial discriminant is just (1−Z₁)², so it vanishes quadratically, not as the square root claimed.
| Claim | Status | What backs it |
|---|---|---|
| The wall can be extrapolated from inside — using only certificates that CLOSED | MUTATION-TESTED | wall_sweep.js, 36 probes. Linear fit of (1−Z₁) over the last four closing points predicts A* at 5.216 / 5.389 / 5.482 for N=10/14/18, each landing in or within 0.02 of its measured bracket. You never have to run past the wall to locate it |
| THE WALL IS NOT A SINGULARITY. σmin(DF) = 0.645 where the certificate stops closing | MUTATION-TESTED | 0.703 at A=4 · 0.645 at A=5.5 · 0.506 at A=9. Falling slowly and linearly; zero extrapolates to A ≈ 22, four times beyond the wall. Identical on 38×38 and 66×66 Jacobians, so converged and grid-independent. The equations are well-conditioned right through the wall |
| So C3's premise, as stated, is dead | OPEN | "the certificate fails because the solution set is changing dimension" is FALSE here. The solution is isolated and well-conditioned at the wall. That reading was what made the wall interesting |
| What the wall actually measures: representation adequacy | STANDARD | Z₁ is a ν-WEIGHTED sequence norm. The crowd concentrates (heatmap: max/min 1.05→2.93), the Fourier tail thickens, the weighted bounds degrade. This explains what the singularity reading could not: why the wall moves with N (more modes), moves with ν (the tail weight), and coexists with a large σmin |
| Two kinds of refusal, and the thesis conflated them | OPEN | GEOMETRIC (Wardrop S1): the Jacobian is genuinely singular along a 6-dim face — there is no isolated point to enclose, and the refusal direction IS the tangent. REPRESENTATION (mfg-congest): the solution is isolated; the basis just cannot represent it. The refusal carries no geometric information. The thesis title is true for the first and false for the second |
| The wall moves with the discretization — and CONVERGES, first order in the grid | MUTATION-TESTED | Extrapolated A* over N = 10, 14, 18, 24, 32, 40: 5.2163 → 5.6863, increments per unit N decaying 0.043 → 0.005. Slopes against 1/N are −6.04, −5.85, −6.84, −6.41, −6.85 — constant, so A*(N) ~ A∞ − c/N with c ≈ 6.4. CORRECTS an earlier entry that said it did not converge: that read three N off 0.2-wide brackets, which cannot resolve a decaying increment |
| …but the LIMIT is not pinned | OPEN | 1/N model gives A∞ ≈ 5.846; geometric extrapolation gives 5.763. They disagree by 0.08. Convergence is evident; the value is a loose bracket ~5.76–5.86 and must never be published as a number |
| The wall is nearly NORM-INDEPENDENT — a plateau across five methods | MUTATION-TESTED | seven Banach weights at fixed N=18: ν = 1.02–1.10 give A* within 0.029 of each other (5.4528–5.4818); beyond ν = 1.15 the wall RETREATS to 5.13. Degradation is downward only — bad methods give smaller walls, never larger, exactly as a sound-but-conservative certificate requires. sup over methods 5.4818 is the sharpest lower bound on the true singularity |
| …and the grid effect is seven times larger than the method effect | MUTATION-TESTED | plateau spread 0.029 against a residual grid effect of 0.2046 (N=18→40) at fixed method. At this resolution the discretization dominates the norm choice by an order of magnitude — so the wall is far more a property of the problem than of the norm |
| But this tested the NORM, not the preconditioner | OPEN | ν is the Banach-algebra weight. The preconditioner — the midpoint inverse — is identical across all seven runs. The row above must be read as norm-independence, and the preconditioner knob is still untested |
| The radii-polynomial discriminant is NOT a sharper observation point than Z₁ = 1 — my own proposal, refuted | MUTATION-TESTED | 4·Z₂·Y₀ measures 3.2e−6, 7.2e−9, 2.7e−11 at the last closing point, so disc = (1−Z₁)² to rounding — the same quantity squared, carrying no extra information. And since (1−Z₁) is LINEAR in A, disc vanishes quadratically, not as the square root I predicted |
| Z₁ = 1 is a property of the METHOD, not of the equations — it depends on the preconditioner, the radius policy, the Newton point | STANDARD | definitional, and raised independently by two of the three attacks. As stated, the wall was not a mathematical object |
| The refusal wall is measured: closes on A ∈ [0.3, 5], refuses at A = 6 (Z₁ = 1.10) | STANDARD | a run record. Sampled, not bracketed — which is precisely what C3 set out to fix |
| Version A — certify the true fold by an extended system in (x, A, v) | OPEN | occupancy HIGH: this is standard validated bifurcation practice. An application, not a method |
| Version B — fix a certificate policy, then bracket Z₁(A) = 1 by interval bisection | OPEN | weaker, but literally what C3 asks. Requires declaring the policy first |
| The distance theorem — Z₁ ≥ f(σmin, Lipschitz), bounding the method's wall against the equations' singularity | OPEN | the real prize, Renegar-style. No occupancy check has been run on it — and that is how the tropical and sheaf proposals died |
| Claim | Status | What backs it |
|---|---|---|
| Three independent attacks agree that C2 is the core | STANDARD | no coordination between them — convergence is the signal |
| Two of three reject “refusal” as a mathematical object | STANDARD | it survives as the way IN, not as the thing certified. The thesis title changes or the first referee does it for us |
| Every position in this document carries a decidable test | MUTATION-TESTED | the record itself — a strategy document that cannot be falsified is a mood board |
| A hybrid none of the three proposed alone is rising | PROSPECTIVE | frame from C, vocabulary from B, falsifier from B, core claim from C. Recorded, not yet executed |
| The programme's value was never novel mathematics | STANDARD | mfg-congest is the first validated ENCLOSURE of known mathematics; wardrop-repro says REPRODUCTION in its title. Ideas are scored on artifact value as well as novelty |
Formalise the Krawczyk breakdown as a theorem, then build a 4-step automated diagnostic engine.
First move: Prove M = I − Y[J] has an eigenvalue ≈ 1 with eigenvector in ker(J(x0)).
Reframe the vocabulary, then get EMPIRICAL evidence before any theorem.
First move: Generate thousands of synthetic systems with KNOWN manifolds and measure angle(predicted tangent, true tangent).
One object unifies all three claims: a verified enclosure of ker(Df). Everything else is classical theory or a corollary.
First move: Check whether S1's cost is AFFINE. If so the equilibrium set is a polyhedral face, exact in rational arithmetic — no intervals at all.
Three readers, no coordination. Convergence here is worth more than any single argument.
C2 is the core and must be attacked first
Unanimous, and it matches the thesis's own self-assessment. The strongest signal in the set — three independent readers converging on the same claim without coordination.
The computational primitive is a verified null-space / singular-value enclosure
A reaches it via interval SVD separation bounds, C via Rump-style verified SVD, B assumes it. Nobody proposes a different primitive. This is what to build first whatever else survives.
The directional split (tangent vs transverse) is the mechanism
A calls it V_∥ / V_⊥; B calls it anisotropic contraction; C calls it Lyapunov–Schmidt. Same object, three vocabularies.
The thesis as currently titled is too broad to defend
B says so explicitly ('avoid making Equilibrium is a geometry the central mathematical claim'); C implies it by narrowing everything to the kernel enclosure. Two of three want the headline changed.
Each divergence is resolved by argument, not preference, and the resolution is dated. Two of these overturn parts of the original thesis.
Status may change only by evidence. Nothing here is claimed.
Read together, the three do not select a winner. They compose into something none of them proposed alone.
Recorded because a comparison that only finds encouragement is worthless.
The headline 'refusal is how you measure it' does not survive as MATHEMATICS. It survives as the way in. Two of three readers rejected it independently and C's objection is referee-grade.
C2's tangency half is classical (constant-rank theorem). The claimable novelty shrinks to one lemma — which is what makes it defensible rather than dismissible.
C1's lower bound is free and UPGRADES Wardrop S1 from DEMONSTRATED to PROVED. That is the cheapest real win available in the whole programme.
C3 is demoted and split; only the distance theorem is worth the name, and it has had no occupancy check.