OpenAI · 114-02 · 2026-09-23 · Theoretical computer science
Approximate counting of common bases of two matroids
preprints/Approximate-counting-of-common-bases-of-two-matroids-September-23-2026
Family 114. Approximate counting of common integer polymatroid bases.
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
Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles. Capacities are binary-encoded, each integer vector counts once, and oracle calls and bit operations outside the oracles are polynomial on every execution. For matroids presented by independence oracles, the results also cover common independent sets of prescribed, unrestricted, or maximum cardinality, even when the ranks differ.
We give a fully polynomial randomized approximation scheme for counting the common bases of two arbitrary matroids of the same rank, supplied by independence oracles. The algorithm requires no explicit representation of either matroid and has polynomial bounds on both oracle calls and bit operations on every execution.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 114: Approximate counting of common integer polymatroid bases. 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.