Cubic border-determinantal lower bound for the permanent
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
Does the border-determinantal complexity of the m×m complex permanent grow at least c $m^{3}$ for one absolute $c>0$ and all sufficiently large m? Border complexity permits coefficient limits of affine-linear determinant representations.Short English statement
Does the border-determinantal complexity of the m×m complex permanent grow at least c $m^{3}$ for one absolute $c>0$ and all sufficiently large m? Border complexity permits coefficient limits of affine-linear determinant representations.Sources
- S1 Original question source or mathematical context. https://arxiv.org/pdf/1604.06431.
- S2 Original question source or mathematical context. https://arxiv.org/html/2202.13016.
- S3 Original question source or mathematical context. https://arxiv.org/abs/2606.13628.
- S4 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for CS-152
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).
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
- OpenAI, A cubic lower bound for border determinantal complexity of the permanent, 2026-09-24; citation key
OAI:A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - Family 108: Lean scope.
- Comparator PermanentCubic.json.
- Comparator PermanentCubic.lean.
- Comparator SmoothInitialForm.json.
- Comparator SmoothInitialForm.lean.
- OpenAI catalogue entry.