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Snaky in 21 Maker moves

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

Settles the Snaky achievement problem: Maker can force the six-cell Snaky shape within 21 of its own moves on the initially empty infinite square board. Maker moves first, each player claims one free cell per turn, and translations, rotations and reflections count as wins. (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

The paper states

Theorem 1

In Snaky in 21 Maker moves

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

In this game, there is one globally legal Maker policy that completes an allowed copy of S within at most 21 actual Maker claims against every legal continuation of Breaker. Equivalently, extend the policy by fresh claims after an earlier win: its Maker set after claim 21 contains a target whenever the first 20 Breaker replies are legal.

Reported proof verification
Three Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
Papers and explanations
One current paper, 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.

Snaky in 21 Maker moves

Undated, current edition.

Read it herePDF

We prove that Maker can achieve the Snaky hexomino within 21 actual Maker moves against arbitrary legal Breaker play on the initially empty infinite square board. The same bound holds on a \(17\times17\) square; in fact, Maker can confine its claims to a fixed 251-cell board.

Three Lean statements in this family

Choose one to see it beside the paper theorem it is paired with.

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.

SnakyCertificate

The paper states

Proposition 10

In Snaky in 21 Maker moves

The 35-move certificate has 616 cards, including its six base cards, and 1,837 placement terms. Every card satisfies A_i^{35}\subsetneq T_i^{35}, o\in T_i^{35}\setminus A_i^{35}, and h_i^{35}\le34. Four of its conditional interfaces are

\begin{array}{c|c|r|r|l} i&A_i^{35}&|T_i^{35}|&h_i^{35}&\text{region disjoint from }T_i^{35}\\ \hline 557&\{(-1,0)\}&118&19&\{(x,y):x\le-3,\ y\le-2\}\\ 595&\{(-1,0)\}&222&27&\{(x,y):x=-2,\ y\le-2\}\\ 603&\{(-1,0)\}&275&27&\{(x,y):x\le-3,\ y=0\}\\ 615&\{(1,0)\}&686&34&\{(1,-1)\}. \end{array}

Each row applies at any Maker turn with its required point owned and its entire envelope free of Breaker, regardless of all other ownership. For row 615 there is also a Breaker-turn conclusion: if Maker owns \{o,(1,0)\} and Breaker avoids T_{615}^{35}, then after any legal next reply Maker wins within 33 further claims, unless already victorious.

The words they share: row, envelope, array, terms, point, disjoint, base, certificate, placement, required, conditional, avoids, four, move, maker, breaker.

Possible Lean counterpart

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

Reconstruction of the 35-move Snaky certificate

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

theorem certificate_correct :    TableValid ∧ placementCount = 1837 ∧    Output 557 (-1, 0) 118 19 {p | p.1 ≤ -3 ∧ p.2 ≤ -2} ∧    Output 595 (-1, 0) 222 27 {p | p.1 = -2 ∧ p.2 ≤ -2} ∧    Output 603 (-1, 0) 275 27 {p | p.1 ≤ -3 ∧ p.2 = 0} ∧    Output 615 (1, 0) 686 34 {(1, -1)} ∧    AllRows := 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.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.
  • Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

SnakyConditional

The paper states

Proposition 10

In Snaky in 21 Maker moves

The 35-move certificate has 616 cards, including its six base cards, and 1,837 placement terms. Every card satisfies A_i^{35}\subsetneq T_i^{35}, o\in T_i^{35}\setminus A_i^{35}, and h_i^{35}\le34. Four of its conditional interfaces are

\begin{array}{c|c|r|r|l} i&A_i^{35}&|T_i^{35}|&h_i^{35}&\text{region disjoint from }T_i^{35}\\ \hline 557&\{(-1,0)\}&118&19&\{(x,y):x\le-3,\ y\le-2\}\\ 595&\{(-1,0)\}&222&27&\{(x,y):x=-2,\ y\le-2\}\\ 603&\{(-1,0)\}&275&27&\{(x,y):x\le-3,\ y=0\}\\ 615&\{(1,0)\}&686&34&\{(1,-1)\}. \end{array}

Each row applies at any Maker turn with its required point owned and its entire envelope free of Breaker, regardless of all other ownership. For row 615 there is also a Breaker-turn conclusion: if Maker owns \{o,(1,0)\} and Breaker avoids T_{615}^{35}, then after any legal next reply Maker wins within 33 further claims, unless already victorious.

The words they share: disjoint, placement, reply, legal, maker, conditional, envelope, avoids, certificate, four, move, required, breaker.

Possible Lean counterpart

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

Conditional winning templates from finite partial positions

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

theorem snaky_winning_strategy_with_legal_states :    ∃ σ : Policy Cell,      (∀ r M B, σ r M B ∉ M ∧ σ r M B ∉ B) ∧      ∀ β : ℕ → Cell, LegalRepliesBeforeFinal σ 35 β ∅ ∅ →        HasSnaky (playState σ 35 β ∅ ∅ 35).1 ∧        (playState σ 35 β ∅ ∅ 35).1.card = 35 ∧        Disjoint (playState σ 35 β ∅ ∅ 35).1 (playState σ 35 β ∅ ∅ 34).2 ∧        ∀ k, k < 35 →          Disjoint (playState σ 35 β ∅ ∅ k).1 (playState σ 35 β ∅ ∅ k).2 ∧          (playState σ 35 β ∅ ∅ k).2.card = k := 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.

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-sandboxed-pass”.

Read the complete Lean statement

SnakyTwentyOne

The paper states

Theorem 1

In Snaky in 21 Maker moves

In this game, there is one globally legal Maker policy that completes an allowed copy of S within at most 21 actual Maker claims against every legal continuation of Breaker. Equivalently, extend the policy by fresh claims after an earlier win: its Maker set after claim 21 contains a target whenever the first 20 Breaker replies are legal.

The words they share: actual, against, breaker, claim, copy, game, legal, maker, within.

Possible Lean counterpart

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

Legal Snaky winning strategy within 21 Maker moves

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

theorem snaky_winning_strategy_21_with_legal_states :    ∃ σ : Policy Cell,      (∀ r M B, σ r M B ∉ M ∧ σ r M B ∉ B) ∧      ∀ β : ℕ → Cell, LegalRepliesBeforeFinal σ 21 β ∅ ∅ →        HasSnaky (playState σ 21 β ∅ ∅ 21).1 ∧        (playState σ 21 β ∅ ∅ 21).1.card = 21 ∧        Disjoint (playState σ 21 β ∅ ∅ 21).1 (playState σ 21 β ∅ ∅ 20).2 ∧        ∀ k, k < 21 →          Disjoint (playState σ 21 β ∅ ∅ k).1 (playState σ 21 β ∅ ∅ k).2 ∧          (playState σ 21 β ∅ ∅ k).2.card = k := 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.

One independent group reports that Comparator accepted a run for this Lean statement file: cert-machine. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.

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: SnakyPrototype.snaky_winning_strategy_21_with_legal_states : ∃ σ : Policy Cell, (∀ r M B, σ r M B ∉ M ∧ σ r M B ∉ B) ∧ ∀ β, LegalRepliesBeforeFinal σ 21 β ∅ ∅ → HasSnaky (playState σ 21 β ∅ ∅ 21).1 ∧...card = 21 ∧... on ℤ×ℤ with all 8 dihedral orientations plus translations, Maker first. The other two challenges are supporting: SnakyCertificate.certificate_correct checks a 610-row certificate table (TableValid ∧ placementCount = 1837 ∧ Output 557...), and SnakyConditional is actually the N=35 strategy from the empty board, NOT the 'four conditional winning templates’ the docs describe (docs/Lean mismatch).

Definitions the group asks readers to compare with the paper: Hand-rolled game model: Policy α := ℕ → Finset α → Finset α → α, playState σ N β M₀ B₀ with Breaker as a fixed sequence β : ℕ → Cell (equivalent to an adaptive Breaker against a deterministic σ) and LegalRepliesBeforeFinal constraining Breaker only for k + 1 < N; looks faithful (final reply unused).

“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

In the Snaky Maker–Breaker game, the players alternately claim cells of $\mathbb Z^2$, and Maker seeks a translated, rotated, or reflected copy of the six-cell Snaky shape. The formalization gives a legal strategy that wins within $21$ actual Maker moves against every legal Breaker play from the empty infinite board. The selected theorem checks disjointness and the move counts throughout play; it does not impose the paper’s smaller finite-board restriction.

The linked supplements reconstruct the older $35$-move appendix’s finite recursive certificate and prove four conditional winning templates from partial positions, assuming the required Maker cells are present and Breaker avoids the corresponding finite envelope.

AI-drafted comparisons, with no recorded human review

  • Reconstruction of the 35-move Snaky certificate SnakyCertificate

    Unreviewed AI draft: states a supporting result. States a reconstruction of the older 35-move appendix certificate, not the 21-move theorem.

  • Conditional winning templates from finite partial positions SnakyConditional

    Unreviewed AI draft: states a supporting result. States conditional winning templates from finite partial positions, supplements to the main strategy rather than the 21-move theorem.

  • Legal Snaky winning strategy within 21 Maker moves SnakyTwentyOne

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: one legal Maker policy completing a copy of Snaky within 21 Maker moves against every legal Breaker continuation on the infinite board. The note says the paper’s additional finite-board restriction is not imposed.

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.

Reconstruction of the 35-move Snaky certificate SnakyCertificate

Three 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-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 1093 s; peak memory 8.7 GiB).11 October 2026Original report

Conditional winning templates from finite partial positions SnakyConditional

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-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Legal Snaky winning strategy within 21 Maker moves SnakyTwentyOne

One report, 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

Other proofs and exact calculations

These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.

  • Theorem 1 and Corollary 6 of Snaky in 21 Maker moves

    Reported by
    cert-machine
    Method
    Exact computation
    Their verdict
    “CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
    What it covers
    “the whole headline (Theorem thm:main, 21 Maker moves on Z^2) and Corollary cor:finite-board, twice: (I) the certificate evaluates to A_727 = {}, h_727 = 21 under the paper’s combination rule — sound only with Lemma lem:combination, proved in prose; (II) a rule-free search shows the certificate-guided policy wins within 21 claims against every Breaker line, given three stated monotonicity/locality reductions”
    Original report
    Their 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.

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

This family’s papers cite no paper elsewhere in the release.

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

No paper elsewhere in the release cites this family’s papers.

Follow-up work by others

  • gotrevor, snaky-15x15 and the playable Snaky site, 8 October 2026. Source

    An independent search found a first-player win for Snaky on 15 by 15; the author reports that OpenAI’s 21-move strategy closed the last three positions the search had not settled, and publishes a Lean proof and a playable site.

  • From community.openai.com, 6 October 2026. Source

    gotrevor reports an independent search-based Snaky 15x15 first-player win with a Lean proof (gotrevor/snaky-15x15).

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 one Lean statement with its paper: SnakyCertificate, SnakyConditional, SnakyTwentyOneSuggest 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 proof and rates its significance, if it holds up, as “Field-specific journal”. The page says that nothing on it is verified.

    Author not stated. Source

  • Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source

    The doc row describes conditional winning templates, but the theorem Comparator checks is the unconditional 35-move win, a different statement.

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
  • 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
  • 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 187 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 187 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form