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CS-123-A1 · revision r1 · on CS-123

OpenAI claim summary for CS-123

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 114; 2026-10-05).

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

Manuscript

Contents
  1. FPRAS for common integer polymatroid bases
  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

FPRAS for common integer polymatroid bases

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 114; 2026-10-05), the asserted result is that common integer polymatroid bases with binary capacities can be counted by an FPRAS in rank-oracle and bit complexity polynomial in the encoded data, without expanding capacities into copies.

Mathematical statement of the claimed result

Manuscript 1.
On $E=[n]$, let $r_1,r_2:2^E\to\mathbb Z_{\ge0}$ be normalized, nondecreasing submodular functions with $r_1(E)=r_2(E)=R$. Exact value oracles and binary $R$ are supplied. Count each integral vector once in \[ \Omega=\{x\in\mathbb Z_{\ge0}^E:x(E)=R,\ x(A)\le r_i(A)\ (A\subseteq E,\ i=1,2)\},\qquad Z=|\Omega|. \] For rational $\varepsilon,\delta\in(0,1)$ the claimed uniform randomized algorithm returns a nonnegative rational $\widehat Z$ with \[ \Pr[(1-\varepsilon)Z\le\widehat Z\le(1+\varepsilon)Z]\ge1-\delta. \] Empty intersections always return zero. Both oracle calls and other bit operations are polynomial in $n$, encoding length $\ell$ of $n,R,\varepsilon,\delta$, $\varepsilon^{-1}$ and $\log\delta^{-1}$ on every execution. Reading answers and writing queries are charged; internal oracle evaluation is not.

Outline of the argument

For manuscript 1 (family 114; 2026-10-05), balanced port profiles encode the count with two auxiliary factorial weights. Their quadratic signatures give a transport variance bound; power annealing, capped restarts, and rational volume estimates implement the ratios with bounded fair-bit work.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 114; 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