we-solve (beta)

CS-152-A1 · revision r1 · on CS-152

OpenAI claim summary for CS-152

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 108; 2026-09-24).

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

Manuscript

Contents
  1. Cubic border-determinantal lower bound for the permanent
  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

Cubic border-determinantal lower bound for the permanent

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 108; 2026-09-24), the asserted result is that complex exact and border determinant representations of the permanent require size of cubic order, with the stated branching-program consequences.

Mathematical statement of the claimed result

Manuscript 1.
Over $\mathbb C$, let $\operatorname{per}_m$ be the permanent of a generic $m\times m$ matrix. Its border determinantal complexity is the least $N$ for which it lies in the coefficientwise closure of determinants of $N\times N$ matrices with affine-linear entries. Exact determinantal complexity omits the closure. The asserted estimates are \[ \overline{\operatorname{dc}}(\operatorname{per}_m)\ge\frac{m^3}{5\,529\,600e},\qquad \operatorname{dc}(\operatorname{per}_m)\ge\frac{m^3}{5\,529\,600e}\quad(m\ge1408). \] The bound is cubic, rather than superpolynomial.

Outline of the argument

For manuscript 1 (family 108; 2026-09-24), smooth initial forms are tested by ambient polars. A characteristic-two adjugate subscheme determines a matrix on a fixed curve; normalization defects and finite-subscheme counts give matrix codimension bounds that persist under coefficientwise determinant limits.

Formal verification

For manuscript 1 (family 108; 2026-09-24), the documented Lean scope is cubic bounds for the permanent in both exact and border determinantal representations, together with the explicit large-size estimate and a smooth-initial-form criterion; the algebraic branching-program applications are outside the selected declarations.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are PermanentCubic.lean, SmoothInitialForm.lean.

Their named declarations include OAI.PermanentBorder.permanent_cubic_lower_bounds, OAI.PermanentBorder.smooth_initial_form_lower_bound.

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