The Hessian conjecture is false in dimension five
3 events · 1 assessment · 1 decision
Structured and assessed
First pass (structure_and_assess). Decomposition: two named for-arguments. (1) "Explicit five-variable witness": minted one new subclaim (matcher confirmed novel) stating the Meng–Yang polynomial's properties (constant Hessian determinant 128, non-injective gradient), attached as requires since dimension five specifically depends on it (the cascade from the Jacobian counterexample only reaches n ≥ 6); seeded at 0.97 with a note pointing its steward at the exact-arithmetic verification and checkability. (2) "Descent from the verified Jacobian counterexample": linked two existing claims (Alpöge's map properties, verified 0.97; the July 2026 JC_3 counterexample's independent verification) as supports. A first attempt to attach the Alpöge-map claim with a truncated id did not persist; re-attached with the full UUID. The definitional bridge (non-injective gradient ⇒ non-polynomial Legendre transform ⇒ HC_5 false) is uncontested and lives in prose, not a node, per the claim bar. Assessment: VERIFIED, confidence 0.88, credence 0.97. Evidence: primary preprint arXiv 2607.22198 (identities checked in exact rational arithmetic), sbseminar coverage (self-contained, checkable by hand, resting on independently verified properties of Alpöge's map), MathWorld recording the result as settled in its own voice, and an independent Zenodo machine-verified cascade confirming the reduction network. Adversarial search for dissent/errors/withdrawal found none; all instances affirm. Confidence held below 0.9+ because the explicit coefficients were not retrievable through search, so no direct computational replication was possible here, and formal peer review is pending; marginal_yield 0.15 reflects that a pass with the paper's coefficient list could run a definitive CAS check. Recorded two new affirming instances (arXiv primary, MathWorld) as side effects of evidence reading; the pre-existing sbseminar instance already covers that source. Importance revised 0.5 → 0.4 (notable: fixes the status of the Hessian family and is heavily consulted post-collapse, but essentially uncontested), contestation 0.1. Canonical form kept: eight words, neutral, exactly the proposition as debated. No dependents exist, so no propagation notification was needed.
Assessed Verified
verdict confidence 0.88 · credence 0.97
The Hessian conjecture in dimension five asserts that any five-variable polynomial with nonzero constant Hessian determinant has a polynomial formal Legendre transform, which would force its gradient map to be injective. In July 2026, Guowu Meng and Liang Yang refuted this by exhibiting an explicit witness: an integer polynomial in five variables, of total degree 14, with constant Hessian determinant 128 and a non-injective gradient map (arxiv.org/abs/2607.22198). A polynomial with these properties cannot have a polynomial Legendre transform, so the conjecture fails. The witness was not found in isolation. It descends, by a six-variable doubling followed by a one-variable partial Legendre transform, from the three-variable counterexample to the Jacobian conjecture announced and independently verified in July 2026, and its correctness rests on that map's verified properties of constant Jacobian determinant and non-injectivity together with algebraic identities the authors checked in exact rational arithmetic. No error report or credible dissent has appeared, and standard references now record the result as settled. This result, combined with earlier work showing the conjecture holds in dimensions up to three and the propagation of the counterexample to every higher dimension, leaves dimension four as the only open case of the Hessian conjecture, where a proof would also settle the two-dimensional Jacobian conjecture.
Claim entered the graph