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Math-2642 · mathematics · Geometry

Liftability without spectrahedrality for a hyperbolicity cone

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Is there a real homogeneous hyperbolic polynomial whose closed cone has an exact finite semidefinite lift but is not itself the positive-semidefinite locus of any affine symmetric pencil in its original variables? Auxiliary variables are allowed for the lift but forbidden in the latter representation.

Short English statement

Is there a real homogeneous hyperbolic polynomial whose closed cone has an exact finite semidefinite lift but is not itself the positive-semidefinite locus of any affine symmetric pencil in its original variables? Auxiliary variables are allowed for the lift but forbidden in the latter representation.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2642-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2642

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).

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

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