A nonspectrahedral hyperbolicity cone

OpenAI's counterexample (2026) to the geometric generalized Lax conjecture: a polynomial of degree 20 in 23 variables, built from the Choi–Lam biquadratic form, is hyperbolic, and no linear matrix inequality of any size describes its hyperbolicity cone. Hyperbolic polynomials and Gårding's convexity theorem, the Choi–Lam form and the absence of dominated bilinear squares, a positive map and the slices of the cone, the obstruction to every pencil through a norm identity and its first variation, and OpenAI's exact lift of the same cone as the projection of a spectrahedron of size 100 with 307 auxiliary variables.

Six lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Hyperbolic polynomials and spectrahedral cones
2. The Choi–Lam form
3. A positive map and a hyperbolic polynomial
4. No linear matrix inequality describes the cone
5. Angular slices of the cone
6. A finite semidefinite lift

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Linear algebra over R and C: determinants, adjugates, eigenvalues and orthogonal diagonalization of real symmetric matrices.
• Derivatives of matrix-valued functions; Rouché's theorem from complex analysis.