The Projected Lax Conjecture: is every hyperbolicity cone a spectrahedral shadow?
A spectrahedral shadow is a set of the form with finitely many real symmetric matrices, that is, a set with a finite semidefinite lift. Netzer and Sanyal (2015) proved that a hyperbolicity cone is a spectrahedral shadow when every nonzero boundary point is a smooth point of its hyperbolic polynomial, and formulated the Projected Lax Conjecture, a weakening of the generalized Lax conjecture. Is every closed hyperbolicity cone a spectrahedral shadow?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Real algebraic geometry; semidefinite lifts of convex cones
- Posed by
- Tim Netzer and Raman Sanyal, Smooth hyperbolicity cones are spectrahedral shadows, Math. Program. 153 (2015)
- Year posed
- 2015
- Years open
- 11y
- Solved
- 2026-10-05
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 35 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
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 via an affine section , 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).
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The construction is an existence argument in high dimension; it relies on a finite-field rank certificate given in an appendix.
Verification
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 , matrix size , ) 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.