We prove that not every hyperbolicity cone is a spectrahedral shadow: some closed hyperbolicity cones admit no finite affine semidefinite lift, even with arbitrary real coefficients and any finite number of auxiliary variables. This disproves the Projected Lax Conjecture and hence the generalized Lax conjecture.

We construct a homogeneous polynomial of degree 16 in 23 real variables whose hyperbolicity cone has no representation by a finite homogeneous real symmetric linear matrix inequality. This disproves the geometric Generalized Lax conjecture.

We construct an exact semidefinite lift of the explicit nonspectrahedral hyperbolicity cone in twenty-three variables defined in the companion paper. The lift is a homogeneous real symmetric pencil of size 100 with 307 auxiliary variables and represents the entire closed cone, including every point with singular X. Thus, although this cone has no semidefinite representation in its original coordinates, it admits one when auxiliary variables are allowed. The stated sizes are not claimed to be minimal.