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Math-2400 · mathematics · Geometry

Spectrahedrality of Hyperbolicity Cones

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $h$ be a real homogeneous polynomial hyperbolic in a direction $e$, meaning that $h(e)>0$ and $t\mapsto h(x-te)$ has only real zeros for every real vector $x$. Must its closed hyperbolicity cone be expressible as $\{x:\sum_i x_iA_i\succeq0\}$ for real symmetric matrices $A_i$? Such a cone is spectrahedral.

Formulation references: [S1], [S2].

Short English statement

Let $h$ be a real homogeneous polynomial hyperbolic in a direction $e$, meaning that $h(e)>0$ and $t\mapsto h(x-te)$ has only real zeros for every real vector $x$. Must its closed hyperbolicity cone be expressible as $\{x:\sum_i x_iA_i\succeq0\}$ for real symmetric matrices $A_i$? Such a cone is spectrahedral.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2400-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2400

The negative manuscript claims a hyperbolicity cone with no finite semidefinite lift. Since every spectrahedral cone has such a lift, this is a counterexample to the universal spectrahedrality question and asserts a stronger obstruction. A separate linked construction gives an exact semidefinite lift for its particular nonspectrahedral cone; a lift with auxiliary variables does not make that cone itself spectrahedral.

Spectrahedrality of Hyperbolicity Cones

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; 2026-09-24), the asserted result is that an explicit degree-sixteen polynomial in twenty-three real variables is hyperbolic but its cone has no finite homogeneous real-symmetric matrix-inequality representation in the original coordinates; In manuscript 2 (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; In manuscript 3 (family 095; 2026-10-05), the asserted result is that some hyperbolicity cones remain outside the class of projections of finite real semidefinite systems, whatever real coefficients and auxiliary variables are used.

Mathematical statement of the claimed result

Manuscript 1.
Use cyclic indices on $y\in\mathbb R^3$ and set $Q(y)_{ii}=y_i^2+y_{i+1}^2$, $Q(y)_{ij}=-y_i y_j$ for $i\ne j$. For real symmetric $4\times4$ matrices, define \[ \Phi_y\begin{pmatrix}a&r^{\mathsf T}\\r&T\end{pmatrix} =\begin{pmatrix}\operatorname{tr}(Q(y)T)&-r^{\mathsf T}Q(y)\\-Q(y)r&aQ(y)\end{pmatrix}, \quad p(X,Z,y)=\det((\det X)Z-\Phi_y(\operatorname{adj}X)). \] The claimed polynomial has degree $20$, satisfies $p(I_4,I_4,0)=1$, and is hyperbolic in that direction. Its closed hyperbolicity cone $K$ admits no direct linear matrix inequality: for every finite $N\ge1$ and every real linear $L$ from the $23$-dimensional ambient space to real symmetric $N\times N$ matrices, \[ K\ne\{(X,Z,y):L(X,Z,y)\succeq0\}. \] This excludes direct spectrahedrality, rather than merely a particular chosen pencil.
Manuscript 2.
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.
Manuscript 3.
The existence claim is a real homogeneous polynomial $p$ and a direction $e$ with $p(e)>0$ such that every univariate $t\mapsto p(te-x)$ is real-rooted, but no finite semidefinite lift represents the resulting closed hyperbolicity cone $K$. Precisely, \[ \nexists N,r<\infty,\ A_i,B_j\in\operatorname{Sym}_N(\mathbb R):\quad K=\{x:\exists y\in\mathbb R^r,\ A_0+\sum_ix_iA_i+\sum_jy_jB_j\succeq0\}. \] This excludes representations even with auxiliary variables, and is stronger than excluding a direct linear matrix inequality.

Outline of the argument

For manuscript 1 (family 095; 2026-09-24), a nonnegative Choi--Lam biquadratic form with no dominated bilinear square is inserted into a determinant construction. Rank-one cone limits force a norm identity under a hypothetical pencil representation, and the resulting tangent obstruction contradicts the scalar form's extremality.

For manuscript 2 (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.

For manuscript 3 (family 095; 2026-10-05), a positive semidefinite Hankel seed with controlled multiplication kernel produces finitely many inverse-limit directions. Separation from Hankel images yields a polynomial patch whose moment deformation contradicts analytic data that any semidefinite lift would require.

Formal verification

For manuscript 1 (family 095; 2026-09-24), the documented Lean scope concerns the main degree-twenty polynomial and the absence of a matrix pencil in its twenty-three coordinates. The paper also reduces that polynomial to degree sixteen while retaining the same cone. This formal scope does not assert that all semidefinite lifts are impossible.

This coverage does not establish a formal proof of the complete problem as stated here.

No Lean formalization is listed at this snapshot for manuscript 2 (family 095; 2026-10-05), manuscript 3 (family 095; 2026-10-05).

The linked family Comparator checks are HyperbolicCones.lean.

Their named declarations include OAI.Paper256.main_result.

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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