Browse › Convex and metric geometry › 095 Hyperbolicity cones without semidefinite lifts
A nonspectrahedral hyperbolicity cone
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Abstract
We construct a homogeneous polynomial of degree 16 in 23 real variables whose hyperbolicity cone has no representation by a finite homogeneous real symmetric linear matrix inequality. This disproves the geometric Generalized Lax conjecture.
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Notes
Follow-up works
- Kevin Shu, A Digestion of A Counterexample to the Generalized Lax Conjecture (blog post, Quasicoherent, 2026-10-12). An exposition of the paper's construction of hyperbolicity cones from quadratic families of positive maps and of why these cones are not spectrahedral.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope note 095: Hyperbolicity cones without semidefinite lifts (lean/docs/095.md, release of 2026-10-06). The release's Lean scope note for the family; it covers this paper's Theorem 1.1, a degree-20 hyperbolic polynomial in 23 variables whose closed cone is not the PSD region of any finite real symmetric linear pencil.
- OpenAI math release, lean/OAI/Analysis/HyperbolicCones/Main.lean (commit adc7f12). Declares
OAI.Paper256.main_resultandcone_not_spectrahedral, stating the dimension, degree, normalization, real-rootedness and non-spectrahedrality claims of Theorem 1.1 (comparator statement in HyperbolicCones.lean); the degree-16 reduced polynomial of Corollary 1.2 is not part of this statement.
Related prior literature mentioned by others
- Victor Vinnikov, LMI Representations of Convex Semialgebraic Sets and Determinantal Representations of Algebraic Hypersurfaces: Past, Present, and Future (arXiv:1205.2286, 2012-05-10). Kevin Shu’s exposition cites this survey as background on earlier constructions of hyperbolic polynomials when introducing the manuscript’s nonspectrahedrality problem.
Possibly relevant literature (found by the agent, not checked)
- Nima Amini, Petter Brändén, Non-representable hyperbolic matroids (arXiv:1512.05878, 2015). Discusses consequences for algebraic versions of the generalized Lax conjecture, bearing on the paper's distinction (Section 1.2) between determinantal representability of a polynomial and spectrahedrality of its cone.
Digests and expositions
- Kevin Shu, A Digestion of A Counterexample to the Generalized Lax Conjecture (Quasicoherent, 2026-10-12). Shu recasts the construction using quadratic families of positive maps and explains the nonspectrahedrality argument through extremal biquadratic forms and differentiated kernel-projection norm identities.
- Latent Seminar, Hyperbolicity cones: matrix descriptions, lifts, and obstructions (2026-10-06). This exposition distinguishes the three manuscripts and outlines the nonspectrahedrality argument, the exact lift, and the separate local obstruction to all finite lifts, while disclaiming independent verification of the complete proofs. [family-level]
- Claude Opus 5.5 / Open Mathematics Courses, A nonspectrahedral hyperbolicity cone (undated). Six lessons with proofs and exercises cover the Choi–Lam construction, the obstruction to spectrahedrality, and the companion exact lift of size 100 with 307 auxiliary variables. [AI-generated per source] [family-level]
News and blogs
- Carlos Toledo, OpenAI’s math release: what it claims beyond what it checks (public audit report, finding G5 dated 2026-10-10). Finding G5 distinguishes the formalized nonspectrahedral cone, which the companion manuscript lifts, from the separate no-lift construction underlying the family’s Projected Lax claim. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 095
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026,
author = {{OpenAI}},
title = {{A nonspectrahedral hyperbolicity cone}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf}{OAI:A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/095.md. The Lean statements have not been checked by people against the paper's claims.
Family 095: Hyperbolicity cones without semidefinite lifts
Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral.
Other papers in this family:
- Hyperbolicity Cones Without Semidefinite Lifts · October 5, 2026
- An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone · October 5, 2026