The generalized Lax conjecture: is every hyperbolicity cone spectrahedral?

Theorem 1.1 (September 24): the degree-20 polynomial $p(X,Z,y)=\det((\det X)Z-\Phi_y(\mathrm{adj}X))$ on $\mathrm{Sym}_4\times\mathrm{Sym}_4\times\mathbb R^3$ is hyperbolic with respect to $(I_4,I_4,0)$, and its closed hyperbolicity cone equals $\{L\succeq0\}$ for no finite real linear pencil $L$. Corollary: the degree-16 quotient $q=p/\det X$ has the same cone. The polynomial is not strictly hyperbolic, consistent with the Kummer-Netzer theorem for strictly hyperbolic polynomials. This cone does have a semidefinite lift with auxiliary variables (companion, October 5); the stronger statement that some hyperbolicity cone has no lift at all is the separate Projected Lax entry.

This result conflicts with a public claim. Gonzalez Nevado's arXiv:2601.12267v1 claims a positive solution of the generalized Lax conjecture; the release's appendix argues that its Theorem 52 is false as stated. Until the conflict is settled in public, the entry is Contested. Theorem 1.1 (September 24): the degree-20 polynomial $p(X,Z,y)=\det((\det X)Z-\Phi_y(\mathrm{adj}X))$ on $\mathrm{Sym}_4\times\mathrm{Sym}_4\times\mathbb R^3$ is hyperbolic with respect to $(I_4,I_4,0)$, and its closed hyperbolicity cone equals $\{L\succeq0\}$ for no finite real linear pencil $L$. Corollary: the degree-16 quotient $q=p/\det X$ has the same cone. The polynomial is not strictly hyperbolic, consistent with the Kummer-Netzer theorem for strictly hyperbolic polynomials. This cone does have a semidefinite lift with auxiliary variables (companion, October 5); the stronger statement that some hyperbolicity cone has no lift at all is the separate Projected Lax entry.