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The paper exhibits an explicit integer polynomial in five variables, of total degree 14 with constant Hessian determinant 128, whose gradient is not injective. Its formal Legendre transform is therefore not a polynomial, so the Hessian conjecture $\mathrm{HC}_5$ is false. The counterexample comes from a one-variable Schur descent applied to the six-variable doubling of Alpöge's 2026 Jacobian counterexample.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.
The disclosure credits the second author with finding the Schur descent of Section 6 with the assistance of ChatGPT Pro, and states that the exact-arithmetic verifications throughout were carried out with assistance from Claude. The descent is the step that produces the counterexample; the surrounding framework is the authors'.
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No independent review, but the result is an explicit polynomial with a stated Hessian determinant, so it is checkable by direct computation. Preprint, not refereed.
Correctness: unknown · statement fidelity: unaudited · peer review: none
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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VibeMath has not independently audited the mathematical statement, proof, or novelty claim.