Browse › Convex and metric geometry › 095 Hyperbolicity cones without semidefinite lifts
An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone
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Abstract
We construct an exact semidefinite lift of the explicit nonspectrahedral hyperbolicity cone in twenty-three variables defined in the companion paper. The lift is a homogeneous real symmetric pencil of size 100 with 307 auxiliary variables and represents the entire closed cone, including every point with singular X. Thus, although this cone has no semidefinite representation in its original coordinates, it admits one when auxiliary variables are allowed. The stated sizes are not claimed to be minimal.
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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, 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)
- James Saunderson, Pablo A. Parrilo, Polynomial-sized Semidefinite Representations of Derivative Relaxations of Spectrahedral Cones (arXiv:1208.1443; Math. Program. 153, 2015). Gives explicit semidefinite lifts, with auxiliary variables, of hyperbolicity cones (derivative relaxations), a precedent for the explicit lift in Theorem 1.1.
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]
- 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
- Emergent Mind: family 095
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026,
author = {{OpenAI}},
title = {{An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026/Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone.pdf}{OAI:An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-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:
- Hyperbolicity Cones Without Semidefinite Lifts · October 5, 2026
- A nonspectrahedral hyperbolicity cone · September 24, 2026 Notes