Lean statement

Conditional winning templates from finite partial positions

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.SnakyConditional.snaky_winning_strategy_with_legal_states

Lean statement file: SnakyConditional.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 187 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

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

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.SnakyConditional.Cell line 5
```
abbrev Cell := ℤ × ℤ
```

definition OAI.SnakyConditional.snaky line 6

def snaky : Finset Cell := {(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1)}

definition OAI.SnakyConditional.orient line 7

def orient (r : Fin 8) (p : Cell) : Cell :=  match r.val with  | 0 => (p.1, p.2)  | 1 => (p.2, p.1)  | 2 => (p.1, -p.2)  | 3 => (-p.2, p.1)  | 4 => (-p.1, p.2)  | 5 => (p.2, -p.1)  | 6 => (-p.1, -p.2)  | _ => (-p.2, -p.1)

definition OAI.SnakyConditional.placement line 17

def placement (r : Fin 8) (t p : Cell) : Cell := orient r p + t

definition OAI.SnakyConditional.HasSnaky line 18

def HasSnaky (M : Finset Cell) : Prop :=  ∃ (r : Fin 8) (t : Cell), snaky.image (placement r t) ⊆ M

abbreviation OAI.SnakyConditional.OrdinaryStrategy.Policy line 25

abbrev Policy (α : Type*) := ℕ → Finset α → Finset α → α

definition OAI.SnakyConditional.OrdinaryStrategy.playState line 26

def playState (σ : Policy α) (N : ℕ) (β : ℕ → α) (M₀ B₀ : Finset α) :    ℕ → Finset α × Finset α  | 0 => (M₀, B₀)  | k + 1 =>      let state := playState σ N β M₀ B₀ k      (insert (σ (N - k) state.1 state.2) state.1, insert (β k) state.2)

definition OAI.SnakyConditional.OrdinaryStrategy.makerAt line 32

def makerAt (σ : Policy α) (N : ℕ) (β : ℕ → α) (M₀ B₀ : Finset α) (k : ℕ) : α :=  let state := playState σ N β M₀ B₀ k  σ (N - k) state.1 state.2

definition OAI.SnakyConditional.OrdinaryStrategy.LegalRepliesBeforeFinal line 35

def LegalRepliesBeforeFinal (σ : Policy α) (N : ℕ) (β : ℕ → α)    (M₀ B₀ : Finset α) : Prop :=  ∀ k, k + 1 < N →    β k ∉ insert (makerAt σ N β M₀ B₀ k) (playState σ N β M₀ B₀ k).1 ∧    β k ∉ (playState σ N β M₀ B₀ k).2

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: disjoint, placement, reply, legal and maker. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: conditional, envelope, avoids, certificate, four, move, required and breaker.

Proposition 10 of Snaky in 21 Maker moves

Proposition

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.

Appendix, The 35-move construction and its conditional interfaces Read it in the paper TeX source, line 150

Show the TeX

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{equation}
\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}
\end{equation}
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.

Other possible paper matches, closest wording first

Theorem 11 of Snaky in 21 Maker moves. Words of the Lean statement, its names split into words, that also appear in the theorem: play, policy, replies, legal and maker. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: required, position and breaker.
Proposition 9 of Snaky in 21 Maker moves. Words of the Lean statement, its names split into words, that also appear in the theorem: strategy, policy, replies, legal and maker. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: move, position and breaker.
Theorem 1 of Snaky in 21 Maker moves. Words of the Lean statement, its names split into words, that also appear in the theorem: policy, replies, legal and maker. Found in the paper’s title, which its main theorem states: snaky. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: breaker.
Proposition 7 of Snaky in 21 Maker moves. Words of the Lean statement, its names split into words, that also appear in the theorem: cells and maker. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: four, finite, position and breaker.
Proposition 5 of Snaky in 21 Maker moves. Words of the Lean statement, its names split into words, that also appear in the theorem: placement. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: appendix and finite.
Corollary 6 of Snaky in 21 Maker moves. Words of the Lean statement, its names split into words, that also appear in the theorem: cell and maker. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite and breaker.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

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

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 187’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

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

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
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).
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“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.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a supporting result”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

56 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 30 to its own declarations, 18 to Mathlib’s documentation. 10 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves snaky_winning_strategy_with_legal_states exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 9 other declarations of the file; with the theorem itself they span 41 lines. 5 of them are named in the statement itself.

Names in the namespace OAI.SnakyConditional are listed without it.

abbrev Cellline 5in the statementlibrary
def snakyline 6library
def orientline 7library
def placementline 17library
def HasSnakyline 18in the statementlibrary
abbrev OrdinaryStrategy.Policyline 25in the statementlibrary
def OrdinaryStrategy.playStateline 26in the statementlibrary
def OrdinaryStrategy.makerAtline 32library
def OrdinaryStrategy.LegalRepliesBeforeFinalline 35in the statementlibrary
theorem snaky_winning_strategy_with_legal_statesline 44selected theoremlibrary

import Mathlibnamespace OAI.SnakyConditionalabbrev Cell := ℤ × ℤdef snaky : Finset Cell := {(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1)}def orient (r : Fin 8) (p : Cell) : Cell :=  match r.val with  | 0 => (p.1, p.2)  | 1 => (p.2, p.1)  | 2 => (p.1, -p.2)  | 3 => (-p.2, p.1)  | 4 => (-p.1, p.2)  | 5 => (p.2, -p.1)  | 6 => (-p.1, -p.2)  | _ => (-p.2, -p.1)def placement (r : Fin 8) (t p : Cell) : Cell := orient r p + tdef HasSnaky (M : Finset Cell) : Prop :=  ∃ (r : Fin 8) (t : Cell), snaky.image (placement r t) ⊆ Mnamespace OrdinaryStrategyvariable {α : Type*} [DecidableEq α]abbrev Policy (α : Type*) := ℕ → Finset α → Finset α → αdef playState (σ : Policy α) (N : ℕ) (β : ℕ → α) (M₀ B₀ : Finset α) :    ℕ → Finset α × Finset α  | 0 => (M₀, B₀)  | k + 1 =>      let state := playState σ N β M₀ B₀ k      (insert (σ (N - k) state.1 state.2) state.1, insert (β k) state.2)def makerAt (σ : Policy α) (N : ℕ) (β : ℕ → α) (M₀ B₀ : Finset α) (k : ℕ) : α :=  let state := playState σ N β M₀ B₀ k  σ (N - k) state.1 state.2def LegalRepliesBeforeFinal (σ : Policy α) (N : ℕ) (β : ℕ → α)    (M₀ B₀ : Finset α) : Prop :=  ∀ k, k + 1 < N →    β k ∉ insert (makerAt σ N β M₀ B₀ k) (playState σ N β M₀ B₀ k).1 ∧    β k ∉ (playState σ N β M₀ B₀ k).2end OrdinaryStrategyopen OrdinaryStrategytheorem 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  sorryend OAI.SnakyConditional

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Snaky in 21 Maker moves

Theorem

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.

Section 1, Introduction Read it in the paper TeX source, line 25

Show the TeX

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.

Configuration and file facts

File: lean/ComparatorChallenges/SnakyConditional.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 187 link it.
Solution module: OAI.GameTheory.SnakyConditional.Main
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 9 local vocabulary declarations
Lines of Lean, without comments or blank lines: 41
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas