OpenAI · 108-01 · 2026-09-24 · Theoretical computer science
A cubic lower bound for border determinantal complexity of the permanent
preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026
Family 108. A cubic permanent–determinant lower bound.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves an lower bound for the border determinantal complexity of the permanent over . Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least , for an absolute and all sufficiently large ; the same bound therefore holds for exact representations.
We prove that the complex border determinantal complexity of the permanent is , allowing arbitrary affine-linear determinant representations and coefficientwise limits. It also gives cubic lower bounds for exact determinantal complexity and for the numbers of vertices and edges in affine-linear algebraic branching programs, including coefficientwise limits with a fixed vertex or edge budget.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 108: A cubic permanent–determinant lower bound. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.