Plectis Atlas
Search

↑ and ↓ move, Enter opens, Esc closes.

Show on the map

Sampling and counting contingency tables with arbitrary margins

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and binary-encoded margins are unrestricted. Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments

What the papers state

Theorem 1.1

In An FPRAS for Cell-Bounded Contingency Tables, 24 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

There is one randomized algorithm with the following guarantee. Given (r,c,b) as above and rational \varepsilon,\delta\in(0,1), it outputs a nonnegative rational \widehat Z such that

\P\bigl((1-\varepsilon)Z(r,c,b)\le \widehat Z \le(1+\varepsilon)Z(r,c,b)\bigr)\ge1-\delta.

If \Omega(r,c,b) is empty, the output is zero on every execution. The algorithm uses independent unbiased bits, and one fixed polynomial in L, \varepsilon^{-1}, and \log(\delta^{-1}) bounds its total bit operations on every execution, where L is the total binary input length, including the accuracy parameters. No counting or sampling oracle is required.

Theorem 1.1

In Exact Uniform Sampling of Contingency Tables with Arbitrary Margins, 24 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

For arbitrary nonnegative integer margins of equal total, the following algorithms exist in the model of unbiased random bits and bit operations.

  1. (i)

    Given an integer k\ge1, a bounded-time algorithm outputs a table in \Omega(r,c) whose law is within 2^{-k} in total variation of the uniform law. Its running time is polynomial in m,n,\log(N+1),k.

  2. (ii)

    An algorithm outputs an exactly uniform table in \Omega(r,c), terminates almost surely, and has expected running time polynomial in m,n,\log(N+1).

The polynomials are uniform over all margin vectors. No positivity, sparsity, balance, or fixed-dimension hypothesis is imposed.

No Lean statement in the release is paired with this theorem.

The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.

Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
Papers and explanations
Two current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Exact Uniform Sampling of Contingency Tables with Arbitrary Margins

24 September 2026, current edition.

Read it herePDF

We give an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed margins. It terminates almost surely and has expected bit complexity polynomial in both dimensions and the binary length of the margins. No positivity, balance, sparsity, or fixed-dimension assumption is required.

An FPRAS for Cell-Bounded Contingency Tables

24 September 2026, current edition.

Read it herePDF

We give a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions vary, all numerical data are encoded in binary, and zero bounds are allowed. The algorithm uses only unbiased random bits and has a polynomial bound on its bit operations on every execution.

One Lean statement in this family

Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

ContingencyTables

The paper states

Theorem 1.1

In An FPRAS for Cell-Bounded Contingency Tables, 24 September 2026

There is one randomized algorithm with the following guarantee. Given (r,c,b) as above and rational \varepsilon,\delta\in(0,1), it outputs a nonnegative rational \widehat Z such that

\P\bigl((1-\varepsilon)Z(r,c,b)\le \widehat Z \le(1+\varepsilon)Z(r,c,b)\bigr)\ge1-\delta.

If \Omega(r,c,b) is empty, the output is zero on every execution. The algorithm uses independent unbiased bits, and one fixed polynomial in L, \varepsilon^{-1}, and \log(\delta^{-1}) bounds its total bit operations on every execution, where L is the total binary input length, including the accuracy parameters. No counting or sampling oracle is required.

The words they share: randomized, including, binary, zero, input, length, rational, execution, accuracy, counting, parameters, one, operations, sampling, algorithm, output, bit, polynomial, bounds, total, fpras, cell, tables.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Exact and bounded-time sampling of contingency tables

ContingencyTables.lean, the release lists it as a supporting statement.

def BoundedSamplingStatement : Prop :=  ∃ (A : Machine) (C d : ℕ), 0 < C ∧    ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),      (∑ i, r i) = ∑ j, c j → ∀ k : ℕ, 1 ≤ k →        let input := encodeSamplingInput r c k        let t := C * (samplingSize n r k) ^ d        (∀ bits : Fin t → Bool, ∃ X : Table r c, OutputsTable A input bits X.val) ∧        (∑ X : Table r c,          |tableMass A input t X.val - (Fintype.card (Table r c) : ℝ)⁻¹|) / 2 ≤            ((2 : ℝ) ^ k)⁻¹theorem boundedSampling : Algorithms.BoundedSamplingStatement := by sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

The file states two more theorems: exactSampling, counting.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-pass”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).

Their note: None material; counting statement assumes equal row/column totals (unequal totals, which also admit no table, are not required to output 0).

Definitions the group asks readers to compare with the paper: Machine model is the hand-rolled RandomMachine (single-tape Post-Turing table over Fin 8, one fair coin per tick) shared with the matching FPRAS file; ‘expected polynomial time’ is tsum (1 - haltMass t), ‘uniform’ is tableMass → 1/card (Table r c), ‘almost uniform’ is total variation (∑ |tableMass - 1/card|)/2; OutputsTable uses an explicit @Fin.instInhabited 8 alphabetSize_neZero instance hack

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).

“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalization gives an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed row and column sums having equal totals. It terminates almost surely, every halted output is feasible, and each table has exactly the uniform limiting probability. Expected bit complexity is polynomial in the dimensions and the binary length of the margins.

It also gives a sampler with a fixed polynomial time bound on every execution whose total-variation error is at most $2^{-k}$ for requested precision $k\ge1$. The same Comparator file includes the companion approximation scheme for counting tables with individual cell bounds. The formalization gives a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions may vary, the margins and bounds are binary encoded, and zero entry bounds are allowed. For rational $0<\varepsilon,\delta<1$, the algorithm returns a nonnegative estimate with relative error at most $\varepsilon$ with probability at least $1-\delta$, and returns zero on every execution when no table exists. The running time is polynomial in the encoded input size, $\varepsilon^{-1}$, and $\log(\delta^{-1})$ on every random tape. The same Comparator file also contains the companion sampling results for tables without cell bounds.

AI-drafted comparisons, with no recorded human review

  • Exact and bounded-time sampling of contingency tables ContingencyTables

    Unreviewed AI draft: states a paper’s theorem as the paper does. The file states the main theorems of both papers: exact uniform sampling of contingency tables with arbitrary margins in polynomial expected bit time with a bounded-time approximate sampler, and the FPRAS for cell-bounded tables.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Exact and bounded-time sampling of contingency tables ContingencyTables

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-pass”Re-run through Comparator: accepted.8 October 2026Original report

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Within this family

Papers elsewhere in the release that this family’s papers cite

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanationCompare the Lean statement with the paperSuggest a source attributionSuggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a algorithm and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 115 No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 115 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form