Disputed

The generalized Lax conjecture: is every hyperbolicity cone spectrahedral?

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.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: contested. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The generalized Lax conjecture: is every hyperbolicity cone spectrahedral? · Generalized Lax conjecture

Confidence: Not scored

Registry verification: contested · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review! Disputed

Named evidence

VibeMathed record: The generalized Lax conjecture: is every hyperbolicity cone spectrahedral?

specialist registry · passed

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.

Source ↗

Companion: A nonspectrahedral hyperbolicity cone

paper · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean proof (OAI.Paper256.main_result)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Comparator statement: HyperbolicCones.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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The Projected Lax Conjecture: is every hyperbolicity cone a spectrahedral shadow?

witness · pending

paper link supplied by the VibeMathed registry.

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Companion: An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone

paper · pending

paper link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean proof (OAI.Paper256.main_result)

lean artifact · pending

Artifact ↗

Comparator statement: HyperbolicCones.lean

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

The Projected Lax Conjecture: is every hyperbolicity cone a spectrahedral shadow? evidence for this event

This event attributed to Unreleased internal OpenAI model

Companion: An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone evidence for this event

Companion: A nonspectrahedral hyperbolicity cone evidence for this event

Lean proof (OAI.Paper256.main_result) evidence for this event

The generalized Lax conjecture: is every hyperbolicity cone spectrahedral? parent of this event

VibeMathed record: The generalized Lax conjecture: is every hyperbolicity cone spectrahedral? evidence for this event

Comparator statement: HyperbolicCones.lean evidence for this event

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