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Hyperbolicity cones without semidefinite lifts

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

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. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments

The paper states

Theorem 1.1

In Hyperbolicity Cones Without Semidefinite Lifts, 5 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Three other possible matches are listed below the card, among them Theorem 1.1 in A nonspectrahedral hyperbolicity cone and Corollary 1.2 in A nonspectrahedral hyperbolicity cone.

There exists a real homogeneous hyperbolic polynomial whose closed hyperbolicity cone has no finite semidefinite lift.

No Lean statement in the release is paired with this theorem.

Three more possible matches

  • Theorem 1.1 in A nonspectrahedral hyperbolicity cone, 24 September 2026.
  • Corollary 1.2 in A nonspectrahedral hyperbolicity cone, 24 September 2026.
  • Theorem 1.1 in An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone, 5 October 2026.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
Papers and explanations
Three current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Hyperbolicity Cones Without Semidefinite Lifts

5 October 2026, current edition.

Read it herePDF

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.

A nonspectrahedral hyperbolicity cone

24 September 2026, current edition.

Read it herePDF

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.

An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone

5 October 2026, current edition.

Read it herePDF

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.

One Lean statement in this family

Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

HyperbolicCones

The paper states

Proposition 4.1

In A nonspectrahedral hyperbolicity cone, 24 September 2026

Let K=\Lambda_+(p,(I_4,I_4,0)) for the polynomial in (1.4). Suppose that K admits a representation by a finite homogeneous real symmetric linear pencil. Then there are integers a,c\geq1, positive linear maps

D:\Sym^4\longrightarrow\Sym^a, \qquad E:\Sym^4\longrightarrow\Sym^c, \qquad D(I_4)=I_a,\quad E(I_4)=I_c,

and a linear map B:\R^3\longrightarrow\R^{a\times c} such that, for every X\succ0, Z\in\Sym^4 and y\in\R^3,

Z\succeq\Phi_y(X^{-1}) \quad\Longleftrightarrow\quad E(Z)\succeq B(y)^\top D(X)^{-1}B(y).

The words they share: integers, symmetric, phi, real, polynomial, homogeneous, sym, pencil, representation, positive, finite, linear.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Nonspectrahedral hyperbolicity cone

HyperbolicCones.lean, the release lists it as a main result.

theorem main_result :    Module.finrank ℝ Ambient = 23 ∧    polynomial.IsHomogeneous 20 ∧    polynomial.totalDegree = 20 ∧    MvPolynomial.eval (coordinates basePoint) polynomial = 1 ∧    (∀ (x : Ambient) (z : ℂ), (linePolynomial x).aeval z = 0 → z.im = 0) ∧    ∀ (N : ℕ) (_hN : 0 < N) (L : Ambient →ₗ[ℝ] Sym N),      cone ≠ {x | (L x : Mat N ℝ).PosSemidef} := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express a weaker statement than the advertised result (their term: “weaker-statement”).

Their note: Only the generalized Lax disproof (cone is not spectrahedral, no auxiliary variables); the Projected Lax disproof (not a spectrahedral shadow, no lift at all) is not stated, and the companion paper shows the formalized cone does have an exact semidefinite lift.

Definitions the group asks readers to compare with the paper: cone := {x | ∀ t : ℂ, (linePolynomial x).aeval t = 0 → t.im = 0 ∧ 0 ≤ t.re} (closed hyperbolicity cone defined via roots of t ↦ p(t e - x), standard) and the polynomial itself via matrixValue/phi/adjugate; only homogeneous pencils L : Ambient →ₗ Sym N are excluded

“weaker-statement”, in their words: The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).

The gap they name: the headline’s lead claim (Projected Lax conjecture false; some hyperbolicity cones are not spectrahedral shadows / admit no lift with auxiliary variables) is not stated; only non-spectrahedrality in the original coordinates is

“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The generalized Lax conjecture predicts that every hyperbolicity cone is spectrahedral. The formalized counterexample is an explicit homogeneous polynomial of degree $20$ in $23$ 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.

AI-drafted comparisons, with no recorded human review

  • Nonspectrahedral hyperbolicity cone HyperbolicCones

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: an explicit degree-20 hyperbolic polynomial in 23 variables whose hyperbolicity cone is not spectrahedral. The semidefinite-lift questions of the family’s later papers are not part of the statement.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Nonspectrahedral hyperbolicity cone HyperbolicCones

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Within this family

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

  • Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source

    The headline claims no exact semidefinite lift, but only non-spectrahedrality is formalised, and a companion paper constructs such a lift for the same cone.

Questions awaiting review

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Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 095 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 095 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form