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Math-2644 · mathematics · Geometry

Projected Lax conjecture

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does every closed hyperbolicity cone $K(p,e)$ of a real homogeneous polynomial hyperbolic in direction $e$ admit a finite exact semidefinite lift? This means \[ K(p,e)=\left\{x:\text{there exists }u\in\mathbb{R}^a\text{ with }A_0+\sum_i x_iA_i+\sum_j u_jB_j\succeq0\right\} \] for some finite $a$ and finite real symmetric matrices, where $\succeq0$ means positive semidefinite; no closure of the projection is allowed.

Short English statement

For every real homogeneous polynomial hyperbolic in direction $e$, is its closed cone $K(p,e)$ exactly the set of $x$ for which some $u\in\mathbb{R}^a$ makes $A_0+\sum_i x_iA_i+\sum_j u_jB_j$ positive semidefinite, for some finite $a$ and finite real symmetric matrices, without taking the closure of the projection?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2644-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2644

The displayed question has a negative answer under the manuscript's stated construction. Its counterexample meets the conditions in this problem and defeats the universal assertion, so this is a full disproof claim. The supporting result is manuscript 1 (family 095; 2026-10-05).

Projected Lax conjecture

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 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.
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; 2026-10-05), 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

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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