| # Hyperbolicity cones without semidefinite lifts | |
| The following describes the scope of the Lean formalization related to the following accompanying paper(s): | |
| - [A nonspectrahedral hyperbolicity cone](../../preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf) | |
| ## Scope | |
| The generalized Lax conjecture predicts that every hyperbolicity cone is spectrahedral. The formalized counterexample is an explicit homogeneous polynomial of degree $20$ in $23$ real variables. It is hyperbolic, but its closed hyperbolicity cone cannot be represented as the positive-semidefinite region of any finite real symmetric linear matrix pencil, in any positive matrix size. | |
| ## Comparator links | |
| | Result | Comparator statement | | |
| | --- | --- | | |
| | Nonspectrahedral hyperbolicity cone | [HyperbolicCones.lean](../ComparatorChallenges/HyperbolicCones.lean) | | |