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Math-2642-A1 · revision r1 · on Math-2642

OpenAI claim summary for Math-2642

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 095; 2026-10-05).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Liftability without spectrahedrality for a hyperbolicity cone
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Liftability without spectrahedrality for a hyperbolicity cone

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 095; 2026-10-05), the asserted result is that the explicit twenty-three-variable nonspectrahedral cone nonetheless has an exact lift with a size-$100$ pencil and $307$ auxiliary variables, including singular boundary points.

Mathematical statement of the claimed result

Manuscript 1.
For cyclic $i\in\{1,2,3\}$ set $Q(y)_{ii}=y_i^2+y_{i+1}^2$, $Q(y)_{ij}=-y_i y_j$ otherwise, and \[ \Phi_y\begin{pmatrix}a&r^{\mathsf T}\\r&T\end{pmatrix} =\begin{pmatrix}\operatorname{tr}(Q(y)T)&-r^{\mathsf T}Q(y)\\-Q(y)r&aQ(y)\end{pmatrix}, \quad p(X,Z,y)=\det((\det X)Z-\Phi_y(\operatorname{adj}X)). \] Let $K$ be the closed hyperbolicity cone of $p$ in direction $(I_4,I_4,0)$ on the real $23$-dimensional space of symmetric $X,Z$ and $y\in\mathbb R^3$. The claim gives fixed real symmetric $100\times100$ matrices $A_1,\ldots,A_{23},B_1,\ldots,B_{307}$ with \[ x\in K\ \Longleftrightarrow\ \exists v\in\mathbb R^{307}:\quad \sum_{\nu=1}^{23}x_{\nu} A_{\nu}+\sum_{j=1}^{307}v_jB_j\succeq0. \] Coordinates list upper-triangular entries of $X$ then $Z$, followed by $y$. Singular boundary points are included. Auxiliary variables make this a lift; it does not assert direct spectrahedrality.

Outline of the argument

For manuscript 1 (family 095; 2026-10-05), the three-variable parameter is covered by rotating two-dimensional planes. Each angular slice has a trigonometric Gram factorization and an exact pencil representation; a finite matching identity combines these slices into the full homogeneous lift.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 095; 2026-10-05).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources