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CS-120-A1 · revision r1 · on CS-120

OpenAI claim summary for CS-120

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 115; 2026-09-24).

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

Manuscript

Contents
  1. Exact uniform sampling of arbitrary-margin contingency tables
  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

Exact uniform sampling of arbitrary-margin contingency tables

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 115; 2026-09-24), the asserted result is that arbitrary-margin nonnegative integer contingency tables admit exact uniform sampling with almost-sure termination and polynomial expected bit cost.

Mathematical statement of the claimed result

Manuscript 1.
For arbitrary nonnegative integer margins $r\in\mathbb Z_{\ge0}^m$, $c\in\mathbb Z_{\ge0}^n$ with equal total $N$, let \[ \Omega(r,c)=\{A\in\mathbb Z_{\ge0}^{m\times n}:\sum_jA_{ij}=r_i,\ \sum_iA_{ij}=c_j\}. \] The claim gives a bounded-time sampler whose output law is within $2^{-k}$ total variation of uniform for every supplied integer $k\ge1$, with cost polynomial in $m,n,\log(N+1),k$. It also gives an exact-uniform sampler terminating almost surely with expected bit time polynomial in $m,n,\log(N+1)$. Both use unbiased bits and are uniform across all margins, including zeros, with no balance or fixed-dimension assumption.

Outline of the argument

For manuscript 1 (family 115; 2026-09-24), small margins are explicit states while large completions are handled by a padded auxiliary chain. Signature and transport bounds give a bounded approximate sampler; tabulation of its dyadic output law enables a rare exhaustive residual draw that corrects the law exactly.

Formal verification

For manuscript 1 (family 115; 2026-09-24), OpenAI lists Lean statements for this assertion: arbitrary-margin nonnegative integer contingency tables admit exact uniform sampling with almost-sure termination and polynomial expected bit cost.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are ContingencyTables.lean.

Their named declarations include OAI.ContingencyTables.boundedSampling, OAI.ContingencyTables.counting, OAI.ContingencyTables.exactSampling.

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