If a vector field is the gradient of anything, under any Riemannian metric, its Jacobian at an equilibrium has real spectrum — a one-page folklore fact, cited below, never claimed here. The contrapositive is checkable by machine: an exact-rational Sturm count that certifies a complex eigenvalue pair rules out every metric and every potential at once. That certificate class appears to be unoccupied — a documented search, not a proof — and this page ships two exhibits of it. The certificate itself runs in this page, on the same spliced kernels the batteries gate.
Suppose V = −gradg E near an equilibrium z*. Then DV(z*) = −G⁻¹H with G positive definite and H symmetric, which is similar to a symmetric matrix — so its spectrum is real. The criterion is folklore (three one-page derivations; the linear-Markov case is a published iff, Dietert 2015, which is strictly stronger). What is checkable, exactly: build the characteristic polynomial of the reduced Jacobian over BigInt rationals, run a Sturm count, and if the count of distinct real roots falls short of the squarefree degree, a complex pair exists — no C¹ metric and no C² potential realize the flow as a gradient near z*. No floating point exists anywhere between the weights and the verdict.
Three boundaries, stated before a referee states them: the statement is local (near this equilibrium); it rules out Riemannian gradient structure only (metriplectic and GENERIC splittings expect a skew part by design and are untouched); and it is the criterion's necessary half — a real spectrum decides nothing in either direction.
Conventional softmax self-attention token dynamics, untied rational weights. At a zero-sum, row-constant-score configuration the softmax collapses to exactly 1/n — the one transcendental object in the problem becomes a rational number — the equilibrium is an identity in ℚ, and the reduced Jacobian is exactly rational. The word for these instances is engineered: over the preregistered enumeration, certification is 26%, not typical, and every outcome ships:
| slice | enumerated | certified | refused | lost |
|---|---|---|---|---|
| n = 2 antipodal, d = 2 | 192 | 0 — and provably always 0 | 192 | 0 |
| n = 2 antipodal, d = 3 | 672 | 224 | 448 | 0 |
The d = 2 zero is a theorem, found by the census and proved the same day: the antipodal linearization is block-circulant and decouples into two operators of dimension d − 1, so at d = 2 both are scalars and the spectrum is real for every admissible weight pair. The battery pins the decoupling exactly — the characteristic polynomial factors bit-for-bit through it. (The claim record's original scope carried the census as a measured negative, "not proved" — written hours before the proof existed; the record now carries a dated amendment rather than a silent edit, so the two documents agree by supersession, not by erasure.)
Degeneracy, flagged not hidden: 296 of the 864 instances carry a repeated eigenvalue (squarefree degree drop), 44 of them among the 224 certified — those pairs live in the squarefree part, which is sound — and the flagship instance below is drop-free, so the exhibit never needs the multiplicity argument.
An honesty note the derivation itself produced: at this equilibrium class the softmax and unnormalized-attention Jacobians differ by a positive scalar only. This page therefore claims non-gradientness of the softmax flow at these equilibria and never claims to separate softmax from unnormalized attention here. The published non-gradient statement (Sinkformers, attributed by the Geshkovski–Letrouit–Polyanskiy–Rigollet survey) is for the Wasserstein-2 metric, in the mean-field limit, under a tied-weight assumption; the exhibit's delta is exactly three axes — metric-universal, finite-n, weight-general.
The Brusselator's equilibrium (A, B/A) is exact for rational parameters with no construction work. That its eigenvalues go complex near the Hopf bifurcation is classical textbook material — the spectrum fact is not claimed. What the exhibit adds is the certified, metric-universal statement of the consequence, in the same machine-checked format: where the pair is certified, no metric and no potential make the flow a gradient near the equilibrium — the instance-wise form of a question the gradient-structure program for reaction networks leaves open beyond detailed balance. And the 2×2 case pays for its smallness: the certificate has an elementary exact twin, sign of (B−1−A²)² − 4A², evaluated beside every verdict. On all 60 grid instances the twin agreed with the kernel, and a deliberately corrupted Jacobian was shown to disagree — the check can go red, and was watched doing it.
The kernels are spliced verbatim from the repository into this page's script — a battery slices the bytes back out and asserts identity with the sources, then re-runs the verdicts from the page's own bytes. This section needs scripts; everything above it is static prose and stands complete without them.
The criterion (folklore; Dietert owns the linear iff; Brooks–Maas characterize metric-existence for a given potential — the delta here is exactly the quantification over the potential, a step their paper does not take); the sentence “attention is not a gradient flow” (in print with an attributed proof); any softmax-vs-unnormalized separation at these equilibria (provably invisible here); anything about trained weights, the mean-field limit, global structure, or metriplectic systems; typicality (the census says 26%, and the word is engineered); any status above conjectured, which is what the ledger derives today.
On the two preregistered enumerations: 224 of 864 attention instances and 37 of 60
Brusselator instances carry an exactly-verified equilibrium whose reduced Jacobian has a
certified non-real pair — hence no C¹ Riemannian metric and no C² potential realize
those flows as gradients near those equilibria; the d = 2 antipodal slice is all-real by a
proved decoupling; and every number regenerates from batteries whose falsifier has a
mutation-verified red. Falsifiers: node research/emergent-geometry/tests/test-instance.js
and node research/emergent-geometry/tests/test-brusselator.js.
The real-spectrum criterion is folklore with a published stronger cousin (Dietert 2015, arXiv:1405.2552 — real diagonalisability, an iff, for linear Markov generators). Metric existence for a given co-vector field is characterized by Brooks & Maas (arXiv:2209.11149) — in their gradient case (Corollary 2.5, read at body level) the linearization must be positive and symmetric with respect to the given potential's Hessian scalar product, which is strictly stronger than the bare real-spectrum condition; the delta here is exactly the quantification over the potential. The non-gradientness of softmax attention is published for one metric: the survey of Geshkovski, Letrouit, Polyanskiy & Rigollet (arXiv:2312.10794, Bull. AMS 2025) attributes to Sinkformers (Sander, Ablin, Blondel & Peyré, arXiv:2110.11773, AISTATS 2022, Proposition 3) the W₂ statement, under their tied-weight Assumption 1. The Brusselator's Hopf analysis is textbook. What was not located anywhere, across two search vocabularies: a machine-checked certificate whose conclusion is the non-existence of gradient structure under every metric — that absence is a documented search recorded in the unit's literature files, not a proof of novelty.
Evidence: one committed claim record per census — attention (with its run record and a falsifier whose mutation was verified red) and Brusselator (with its run record) · preregistration research/emergent-geometry/PREREG.md (+ dated amendment A1, committed before its code) · derivation and lemma research/emergent-geometry/docs/INSTANCE_DERIVATION.md.
Reproduce: make check-emergent · page battery: node research/emergent-geometry/tests/test-ng-page.js. Paths are monorepo paths; this unit's export decision is recorded in its release record.