The generalized Lax conjecture: is every hyperbolicity cone spectrahedral?

A homogeneous real polynomial $p$ is hyperbolic with respect to $e$ if $p(e)\ne0$ and $t\mapsto p(te-x)$ has only real roots for every $x$ ; its closed hyperbolicity cone is the set of $x$ for which all these roots are nonnegative, a convex cone by Garding. Lax (1958) asked about three variables, and Helton-Vinnikov with Lewis-Parrilo-Ramana showed that every ternary hyperbolic polynomial has a definite determinantal representation. Branden showed that in more variables a hyperbolic polynomial and its powers can fail to be determinantal, which leaves the geometric question about the cone. Is every hyperbolicity cone spectrahedral, that is, of the form $\{x: \sum_i x_iA_i\succeq0\}$ for finitely many real symmetric matrices $A_i$ ?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Real algebraic geometry; hyperbolic polynomials, spectrahedra
Posed by: Generalization of P. D. Lax's 1958 three-variable conjecture; the manuscripts do not name who first stated the general form
Year posed: —
Years open: —
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Contested
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 42 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1 (September 24): the degree-20 polynomial $p(X,Z,y)=\det((\det X)Z-\Phi_y(\mathrm{adj}X))$ on $\mathrm{Sym}_4\times\mathrm{Sym}_4\times\mathbb R^3$ is hyperbolic with respect to $(I_4,I_4,0)$ , and its closed hyperbolicity cone equals $\{L\succeq0\}$ for no finite real linear pencil $L$ . Corollary: the degree-16 quotient $q=p/\det X$ has the same cone. The polynomial is not strictly hyperbolic, consistent with the Kummer-Netzer theorem for strictly hyperbolic polynomials. This cone does have a semidefinite lift with auxiliary variables (companion, October 5); the stronger statement that some hyperbolicity cone has no lift at all is the separate Projected Lax entry.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family's later manuscript (October 5, 2026) disproves the stronger Projected Lax Conjecture by a different, nonexplicit construction, and a third (October 5) builds an exact semidefinite lift of the explicit cone given here.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the September 24 manuscript was read against the conjecture as stated there; it gives an explicit polynomial $p(X,Z,y)=\det((\det X)Z-\Phi_y(\mathrm{adj}X))$ of degree 20 in 23 variables, built from the Choi-Lam biquadratic form, whose cone is not cut out by any finite homogeneous symmetric pencil (the abstract's degree 16 refers to the reduced quotient $p/\det X$ , same cone). The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator HyperbolicCones, declaration OAI.Paper256.main_result). ComparatorChallenges/HyperbolicCones.lean was read here: it defines this polynomial on $\mathrm{Sym}_4\times\mathrm{Sym}_4\times\mathbb R^3$ , asserts dimension 23, homogeneity of degree 20, $p(e)=1$ , real-rootedness along every line, and that for every $N>0$ and every linear map $L$ to $N\times N$ symmetric matrices the cone differs from $\{x:L(x)\succeq0\}$ . This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The manuscript's appendix argues that Theorem 52 of Gonzalez Nevado's arXiv 2601.12267v1, which claims a positive solution, is false as stated. Listed as Contested because of the conflicting claim described in the claim issue.

Claim issue

This result conflicts with a public claim. Gonzalez Nevado's arXiv:2601.12267v1 claims a positive solution of the generalized Lax conjecture; the release's appendix argues that its Theorem 52 is false as stated. Until the conflict is settled in public, the entry is Contested.

The generalized Lax conjecture: is every hyperbolicity cone spectrahedral?

What was actually shown

What the AI did

Verification

Claim issue

Sources

Discussion