What was already known
The three-variable Lax conjecture was proved by Helton and Vinnikov. In higher dimension, Brändén ruled out determinant representations for powers of a particular polynomial, but that did not rule out a different determinant defining the same cone; Kummer exhibited this distinction (companion manuscript introduction, Historical context, p. 2).
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Disproves the geometric Generalized Lax conjecture that every hyperbolicity cone is spectrahedral.
Limitations
The selected cone is nonspectrahedral in its original variables. This does not imply it lacks an auxiliary-variable semidefinite lift; the family’s stronger projected-Lax counterexample is a separate unformalized claim. Theorem 1.1 uses a degree-20 polynomial; its degree-16 quotient appears in Corollary 1.2.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note matches Theorem 1.1 of the companion: an explicit degree-20 hyperbolic polynomial in 23 variables whose cone has no spectrahedral representation.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- The explicit degree-20 homogeneous polynomial in 23 real variables is hyperbolic, and its closed hyperbolicity cone is not the positive-semidefinite region of any finite symmetric linear matrix pencil.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary companion manuscript, attribution, prior-work, and exact Lean-scope review for non-notifying local inclusion.
Read this published revision