Result 095, Convex and metric geometry

Hyperbolicity cones without semidefinite lifts

Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Adding extra variables can turn a difficult convex region into one described by matrix inequalities. The manuscript reports that some hyperbolicity cones resist every finite enlargement, setting a limit on this strategy for representing convex sets.

What changes?

Hyperbolicity cones are convex regions associated with homogeneous polynomials that have only real roots along lines in a chosen direction. A semidefinite lift describes a region using auxiliary variables and an affine symmetric matrix required to have nonnegative eigenvalues. The manuscript claims that some closed hyperbolicity cones admit no exact finite lift, regardless of the real coefficients or number of auxiliary variables. This would disprove the Projected Lax conjecture and, consequently, the generalized Lax conjecture.

What does that help mathematicians do?

Failure in the original coordinates is not enough to establish this barrier. The companion abstracts describe a degree-16 polynomial in 23 variables whose cone lacks a finite homogeneous symmetric matrix-inequality representation, yet admits an exact lift with matrix size 100 and 307 auxiliary variables. It covers the entire closed cone, including singular points; these sizes are not claimed minimal. Thus that example is not itself evidence against lifts: the main claim needs a stronger obstruction.

Are there practical applications?

The immediate value is foundational for convex optimization: the reported counterexamples rule out a universal exact conversion of hyperbolicity-cone constraints into finite semidefinite constraints, even with unrestricted auxiliary variables. This limits a representation strategy, not every optimization method. The claim does not rule out useful approximations or exact lifts for particular cones.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Hyperbolicity Cones Without Semidefinite Lifts

October 5, 2026 29 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026,
  author = {{OpenAI}},
  title = {{Hyperbolicity Cones Without Semidefinite Lifts}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026/nonliftable-hyperbolicity.pdf}{OAI:Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026}},
  year = {2026}
}

A nonspectrahedral hyperbolicity cone

September 24, 2026 14 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026,
  author = {{OpenAI}},
  title = {{A nonspectrahedral hyperbolicity cone}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf}{OAI:A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026}},
  year = {2026}
}

An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone

October 5, 2026 10 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026,
  author = {{OpenAI}},
  title = {{An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026/Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone.pdf}{OAI:An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/095.md.

Hyperbolicity cones without semidefinite lifts

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The generalized Lax conjecture predicts that every hyperbolicity cone is spectrahedral. The formalized counterexample is an explicit homogeneous polynomial of degree 2020 in 2323 real variables. It is hyperbolic, but its closed hyperbolicity cone cannot be represented as the positive-semidefinite region of any finite real symmetric linear matrix pencil, in any positive matrix size.

Comparator links

Result Comparator statement
Nonspectrahedral hyperbolicity cone HyperbolicCones.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.