OpenAI · 095-01 · 2026-10-05 · Convex and metric geometry
Hyperbolicity Cones Without Semidefinite Lifts
preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026
Family 095. Hyperbolicity cones without semidefinite lifts.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
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.
Preprint source and Lean scope
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A Lean scope note is stored for family 095, but it does not name this preprint. It names 095-02. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.