an explicit counterexample to Zhao's Vanishing Conjecture

Authorship disclaimer: This proof was written primarily by GPT-5.6-sol.

on july 19, 2026, levent alpöge posted an explicit counterexample to the jacobian conjecture in three variables. the map has constant nonzero jacobian determinant and sends three different points to the same place. the construction has also been checked in lean.

so the jacobian conjecture is false in every dimension at least three. the two-variable case is still open.

i asked codex to go counterexample hunting. it did not magically break ryser, seymour's second-neighborhood conjecture, or frankl's union-closed sets conjecture. it did find a more grounded casualty of the jacobian counterexample: an explicit homogeneous quartic that refutes Zhao's Vanishing Conjecture.

the result

there is an explicit homogeneous quartic

P in Q(i)[A1,...,A23,B1,...,B23]

with 410 terms such that:

• Hess(P) is nilpotent;
• the map Z - grad(P) sends three distinct exact points to the same image;
• Delta^m(P^m) = 0 for every positive m; and
• Delta^m(P^(m+1)) is nonzero for infinitely many m.

the last two lines say exactly that P is a counterexample to the homogeneous quartic Vanishing Conjecture in Zhao's formulation.

this is not only an existence argument. the polynomial, all three collision points, and their common image are serialized over Q(i) in a machine-readable certificate. the verifier reconstructs the 410-term polynomial, differentiates it, and checks the collision with exact arithmetic.

where it comes from

the extraction is a chain of old reductions applied to the new map:

1. start with a public 11-variable degree-at-most-three stable lift of alpöge's map, posted by Spacerat and credited there to chatgpt. its full symbolic jacobian determinant is -2.
2. normalize the linear part and apply the Bass-Connell-Wright reduction. this produces a cubic-homogeneous map Id + H in 23 variables with nilpotent JH and an exact three-point collision.
3. apply the de Bondt-van den Essen symmetric lift. this produces the quartic P in 46 variables with nilpotent hessian and carries the collision to Id - grad(P).
4. Zhao's inversion formula turns eventual vanishing of Delta^m(P^(m+1)) into a polynomial inverse for Id - grad(P). the collision makes that impossible.

the verifier checks the original determinant, the normalization and homogeneous decomposition, the block identities behind the 23-variable nilpotence argument, and every collision. the 46-variable hessian step and the laplacian consequences use the cited published theorems rather than an expanded symbolic matrix power.

code and certificate

• self-contained source and proof sketch
• exact 410-term certificate
• verification script

the certificate's sha-256 is b719b1a64d83b96c19455e4292c9f02778335374632bce8d7dcb9bd7b686dfd2. four exact tests pass.

epistemic status

this is a conventional computer-assisted proof built from exact finite checks and published reduction theorems. multiple independent audit passes found no remaining algebraic or sign error; one of them caught an important sign before publication.

it has not been peer reviewed, and I am not claiming priority. Z. Zhang's july 20 note already records the failure of Zhao's conjecture as an existential consequence of the new jacobian counterexample and asks for a small explicit quartic. this is one concrete extraction: 46 variables, 410 terms, exact collision included.