Browse › Convex and metric geometry › 095 Hyperbolicity cones without semidefinite lifts
Hyperbolicity Cones Without Semidefinite Lifts
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Abstract
We prove that not every hyperbolicity cone is a spectrahedral shadow: some closed hyperbolicity cones admit no finite affine semidefinite lift, even with arbitrary real coefficients and any finite number of auxiliary variables. This disproves the Projected Lax Conjecture and hence the generalized Lax conjecture.
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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, Hyperbolicity cones without semidefinite lifts — Lean scope note (OpenAI Math Release, 2026). The family scope note identifies the degree-20 polynomial in 23 variables and nonspectrahedrality of its cone from the September manuscript as the selected formalized result. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Hamza Fawzi, The set of separable states has no finite semidefinite representation except in dimension 3×2 (arXiv:1905.02575, 2019). Another proof that a convex semialgebraic set has no finite semidefinite lift, using elementary semialgebraic arguments; it may be compared with the obstruction in Section 5, which uses semialgebraic selection and analytic lift coordinates.
- Prasad Raghavendra, Nick Ryder, Nikhil Srivastava, Benjamin Weitz, Exponential lower bounds on spectrahedral representations of hyperbolicity cones (arXiv:1711.11497, 2017; SODA 2019). Gives size lower bounds for spectrahedral representations of hyperbolicity cones, the quantitative counterpart that the paper says its existence result does not provide.
Digests and expositions
- 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]
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:Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026,
author = {{OpenAI}},
title = {{Hyperbolicity Cones Without Semidefinite Lifts}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026/nonliftable-hyperbolicity.pdf}{OAI:Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026}},
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:
- A nonspectrahedral hyperbolicity cone · September 24, 2026 Notes
- An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone · October 5, 2026