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. (Lean)

Manuscripts (3)

Hyperbolicity Cones Without Semidefinite Lifts

October 5, 2026 · source · cite as \cite{OAI:Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026}

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.

A nonspectrahedral hyperbolicity cone

September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026}

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.

An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone

October 5, 2026 · source · cite as \cite{OAI:An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026}

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.

Result 095 · Convex and metric geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.