The Projected Lax Conjecture: is every hyperbolicity cone a spectrahedral shadow?

Theorem 1.1: there exists a real homogeneous hyperbolic polynomial whose closed hyperbolicity cone admits no finite semidefinite lift (exact equality, no closure). The cone comes from a positive semidefinite matrix-valued quadratic map $Q$ via an affine section $E_Q=\{(A,t,y):A\succeq0,\ t\ge\mathrm{tr}(A^\dagger Q(y))\}$, and a local sums-of-squares obstruction in the spirit of Scheiderer rules out every lift. This also disproves the generalized Lax conjecture. It is an existence result in high dimension, not an explicit polynomial and not a lower bound on lift size. Note the explicit cone of the September 24 companion does have an exact lift (size 100 pencil, 307 auxiliary variables).

Theorem 1.1: there exists a real homogeneous hyperbolic polynomial whose closed hyperbolicity cone admits no finite semidefinite lift (exact equality, no closure). The cone comes from a positive semidefinite matrix-valued quadratic map $Q$ via an affine section $E_Q=\{(A,t,y):A\succeq0,\ t\ge\mathrm{tr}(A^\dagger Q(y))\}$, and a local sums-of-squares obstruction in the spirit of Scheiderer rules out every lift. This also disproves the generalized Lax conjecture. It is an existence result in high dimension, not an explicit polynomial and not a lower bound on lift size. Note the explicit cone of the September 24 companion does have an exact lift (size 100 pencil, 307 auxiliary variables).