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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).

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Oct 5, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

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

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the October 5 manuscript was read against the conjecture as the paper attributes it to Netzer and Sanyal; it claims a real homogeneous hyperbolic polynomial whose closed hyperbolicity cone has no finite affine semidefinite lift, for any number of auxiliary variables and arbitrary real coefficients. The proof was not refereed. The construction is nonexplicit (quadratic map with $m=330$ , matrix size $20N-1$ , $N>10\cdot36^2$ ) and uses a seed Hankel matrix established by a finite-field certificate in an appendix, which was not re-run here. This manuscript has no Lean formalization in the release; the family's Lean statement covers only the weaker spectrahedrality claim of the September 24 companion.