Lean statement

Finite counterexample to Truffet’s optimization procedure

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.

Finite counterexample to Truffet's optimization procedure

Theorem selected for verification: OAI.TruffetV4Counterexample.finite_execution_not_optimal

Lean statement file: TruffetCounterexample.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 104 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 finite_execution_not_optimal :    crossRowRestriction initial ∧    ForcedSingletonStep initial middle 0 1 ∧    ForcedSingletonStep middle finalState 1 0 ∧    (¬ ∃ j, finalState.remaining j) ∧ stop finalState ∧    feasible initial output ∧ feasible initial better ∧    output 2 = 0 ∧ better 2 = 0 ∧    (output 1 : Coeff) = eval (lowerForm initial 0 1) output ∧    (output 0 : Coeff) = eval (lowerForm middle 1 0) output ∧    eval initial.cost output = eval finalState.cost output ∧    eval initial.cost better < eval finalState.cost output := 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.TruffetV4Counterexample.Coeff line 8
```
abbrev Coeff := WithBot ℝ
```
abbreviation OAI.TruffetV4Counterexample.Form line 10
```
abbrev Form := Fin 3 → Coeff
```
abbreviation OAI.TruffetV4Counterexample.Var line 12
```
abbrev Var := Fin 2
```
abbreviation OAI.TruffetV4Counterexample.RowIndex line 14
```
abbrev RowIndex := Fin 2
```

definition OAI.TruffetV4Counterexample.column line 16

def column (j : Var) : Fin 3 := ⟨j.val, Nat.lt_trans j.isLt (by decide)⟩

definition OAI.TruffetV4Counterexample.nullForm line 18
```
def nullForm : Form := fun _ => ⊥
```

structure OAI.TruffetV4Counterexample.Row line 20

structure Row where  lhs : Form  rhs : Form

structure OAI.TruffetV4Counterexample.State line 24

structure State where  rows : RowIndex → Row  cost : Form  remaining : Var → Prop

definition OAI.TruffetV4Counterexample.wellFormed line 29

def wellFormed (s : State) : Prop := ∀ j, ¬ s.remaining j →  s.cost (column j) = ⊥ ∧ ∀ i,    (s.rows i).lhs (column j) = ⊥ ∧ (s.rows i).rhs (column j) = ⊥

definition OAI.TruffetV4Counterexample.eval line 33

def eval (f : Form) (w : Fin 3 → ℝ) : Coeff :=  max (f 0 + (w 0 : Coeff)) (max (f 1 + (w 1 : Coeff)) (f 2 + (w 2 : Coeff)))

definition OAI.TruffetV4Counterexample.feasible line 36

def feasible (s : State) (w : Fin 3 → ℝ) : Prop :=  ∀ i, eval (s.rows i).rhs w ≤ eval (s.rows i).lhs w

definition OAI.TruffetV4Counterexample.lower line 39

def lower (s : State) (i : RowIndex) (j : Var) : Prop :=  (s.rows i).lhs (column j) ≠ ⊥ ∧    (s.rows i).rhs (column j) < (s.rows i).lhs (column j)

definition OAI.TruffetV4Counterexample.upper line 43

def upper (s : State) (i : RowIndex) (j : Var) : Prop :=  (s.rows i).rhs (column j) ≠ ⊥ ∧    (s.rows i).lhs (column j) < (s.rows i).rhs (column j)

definition OAI.TruffetV4Counterexample.dominating line 47

def dominating (s : State) (j : Var) : Prop :=  s.remaining j ∧ (∃ i, lower s i j) ∧ ∃ i, upper s i j

definition OAI.TruffetV4Counterexample.eligible line 50

def eligible (s : State) (i : RowIndex) (j : Var) : Prop :=  s.remaining j ∧ lower s i j ∧ ((∃ k, dominating s k) → dominating s j)

definition OAI.TruffetV4Counterexample.bounded line 53

def bounded (s : State) : Prop :=  ∀ j, s.remaining j → (∃ i, lower s i j) ∨ ∃ i, upper s i j

definition OAI.TruffetV4Counterexample.stop line 56

def stop (s : State) : Prop := ∀ i, (s.rows i).rhs 2 ≤ (s.rows i).lhs 2

definition OAI.TruffetV4Counterexample.switch line 58

def switch (s : State) : Prop :=  (∀ j, s.remaining j → s.cost (column j) = ⊥) ∧ ¬ stop s ∧  (∃ j, s.remaining j) ∧  (¬ ∃ i j, s.remaining j ∧ lower s i j) ∧  ∃ i j, s.remaining j ∧ upper s i j

definition OAI.TruffetV4Counterexample.failure line 64

def failure (s : State) : Prop := ¬ stop s ∧  ((¬ ∃ j, s.remaining j) ∨   ((¬ ∃ i j, s.remaining j ∧ lower s i j) ∧    (¬ ∃ i j, s.remaining j ∧ upper s i j)))

definition OAI.TruffetV4Counterexample.lowerForm line 69

def lowerForm (s : State) (i : RowIndex) (j : Var) : Form := fun k =>  if k = column j then ⊥ else    (s.rows i).rhs k + ((-WithBot.unbotD 0 ((s.rows i).lhs (column j)) : ℝ) : Coeff)

definition OAI.TruffetV4Counterexample.substitute line 73

def substitute (f : Form) (j : Var) (g : Form) : Form := fun k =>  if k = column j then ⊥ else max (f k) (f (column j) + g k)

definition OAI.TruffetV4Counterexample.cleanup line 76

def cleanup (r : Row) : Row := by  classical  exact if ∀ k, r.rhs k ≤ r.lhs k then ⟨nullForm, nullForm⟩ else r

definition OAI.TruffetV4Counterexample.advance line 80

def advance (s : State) (i : RowIndex) (j : Var) : State where  rows := fun r => cleanup ⟨substitute (s.rows r).lhs j (lowerForm s i j),    substitute (s.rows r).rhs j (lowerForm s i j)⟩  cost := substitute s.cost j (lowerForm s i j)  remaining := fun k => s.remaining k ∧ k ≠ j

definition OAI.TruffetV4Counterexample.ForcedSingletonStep line 86

def ForcedSingletonStep (s s' : State) (i : RowIndex) (j : Var) : Prop :=  wellFormed s ∧ wellFormed s' ∧  bounded s ∧ ¬ stop s ∧ ¬ switch s ∧ ¬ failure s ∧  (∀ r k, eligible s r k ↔ r = i ∧ k = j) ∧  (∃ k, (substitute s.cost j (lowerForm s i j)) k ≠ ⊥) ∧  s' = advance s i j

6 more definitions are reached the same way; the full file lists them.

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: remains, stated, lower and output.

Proposition 5.1 of Randomized quasipolynomial-time mean-payoff games

Proposition (Orientation)

Assume the lower-rank comparison guarantees stated at the beginning of this section. On a fixed permitted input (a,b;B,\alpha,\beta,k,m) of rank r, let (\chi,\mathcal H) be the output of \Orient. For every fixed admissible subsolution y of [a,b] whose core in Equation (eq:orientation-cores) satisfies \alpha(I_y)>\eta k,

\Pr\left( \chi=Y \ \text{or}\ \exists H\in\mathcal H: \alpha(S_y\cap H)>\eta k \right)\geq 1-\frac1{16}.

Separately, for every fixed admissible supersolution z of [a,b] with \beta(I_z)>\eta m,

\Pr\left( \chi=Z \ \text{or}\ \exists H\in\mathcal H: \beta(S_z\cap H)>\eta m \right)\geq 1-\frac1{16}.

On every tape, every set in \mathcal H has \mu-measure at most 1/2. Each probability statement remains valid conditional on a pre-call history that fixes its input and qualifying comparison, when the call uses fresh random bits.

Section 5, The orientation guarantee Read it in the paper TeX source, line 38

Show the TeX

Assume the lower-rank comparison guarantees stated at the beginning of
this section. On a fixed permitted input
\((a,b;B,\alpha,\beta,k,m)\) of rank \(r\), let
\((\chi,\mathcal H)\) be the output of \(\Orient\).
For every fixed admissible subsolution \(y\) of \([a,b]\) whose core
in Equation~\eqref{eq:orientation-cores} satisfies
\(\alpha(I_y)>\eta k\),
\begin{equation}

  \Pr\left(
     \chi=Y
     \ \text{or}\
     \exists H\in\mathcal H:
          \alpha(S_y\cap H)>\eta k
  \right)\geq 1-\frac1{16}.
\end{equation}
Separately, for every fixed admissible supersolution \(z\) of
\([a,b]\) with \(\beta(I_z)>\eta m\),
\begin{equation}

  \Pr\left(
     \chi=Z
     \ \text{or}\
     \exists H\in\mathcal H:
          \beta(S_z\cap H)>\eta m
  \right)\geq 1-\frac1{16}.
\end{equation}
On every tape, every set in \(\mathcal H\) has \(\mu\)-measure at
most \(1/2\). Each probability statement remains valid conditional
on a pre-call history that fixes its input and qualifying comparison,
when the call uses fresh random bits.

Other possible paper matches, closest wording first

Corollary 8.1 of Randomized quasipolynomial-time mean-payoff games. Words of the Lean statement, its names split into words, that also appear in the theorem: failure, upper, cost and max. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: terminates.
Corollary 9.1 of Deterministic quasipolynomial-time mean-payoff games. Words of the Lean statement, its names split into words, that also appear in the theorem: real, row, finite, bound and max. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two.
Corollary 7.2 of Deterministic quasipolynomial-time mean-payoff games. Words of the Lean statement, its names split into words, that also appear in the theorem: optimal and output. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: specified.
Theorem 2.4 of Deterministic quasipolynomial-time mean-payoff games. Words of the Lean statement, its names split into words, that also appear in the theorem: index and max. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: procedure and terminates.
Corollary 8.1 of Deterministic quasipolynomial-time mean-payoff games. Words of the Lean statement, its names split into words, that also appear in the theorem: form, finite, bound and max.
Theorem 7.1 of Randomized quasipolynomial-time mean-payoff games. Words of the Lean statement, its names split into words, that also appear in the theorem: execution and bound.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 83 s).
Reported 10 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, Classical.choice, Quot.sound

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 104’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “weaker-statement”.
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement. Their record
Their note: Only a randomized (error 1/8) quasipolynomial algorithm for the winning set; the deterministic algorithm, exact values/optimal positional strategies, turn-based stochastic games and mean-payoff-parity games (two of four index papers) are not stated. TruffetCounterexample is a side negative result.
Definitions the group asks readers to compare with the paper: Custom 3-tape randomized Turing machine (structure Machine, step, run M code random, one random bit per step, alphabet Option Bool, heads in ℤ) with T existentially quantified per input and time = number of steps; standard in form but NOT Mathlib's TM model, so 'quasipolynomial bit time' rests on this hand-rolled definition. Mean payoff is liminf of running averages and strategies are history-dependent (Strategy G := List (Fin G.m) → ...), both standard
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNDECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so. Their record
Their note: Headline: deterministic quasipolynomial algorithms that compute values and optimal positional strategies (ordinary games), the nonnegative-value set of stochastic games, and mean-payoff-parity winning sets. Lean: one randomized (≥ 7/8 success) quasipolynomial algorithm for the zero-threshold winning set of ordinary games, plus a finite counterexample to Truffet's procedure. Of the four manuscripts, only the randomized one is formalized.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization checks that a specified two-step execution of Truffet's elimination procedure terminates with a feasible but nonoptimal output. It is a finite counterexample to that proposed optimization step.”
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

134 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: 162 to its own declarations, 17 to Mathlib’s documentation. 31 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 finite_execution_not_optimal 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 30 other declarations of the file; with the theorem itself they span 92 lines. 13 of them are named in the statement itself.

Names in the namespace OAI.TruffetV4Counterexample are listed without it.

abbrev Coeffline 8in the statementlibrary
abbrev Formline 10library
abbrev Varline 12library
abbrev RowIndexline 14library
def columnline 16library
def nullFormline 18library
structure Rowline 20library
structure Stateline 24in the statementlibrary
def wellFormedline 29library
def evalline 33in the statementlibrary
def feasibleline 36in the statementlibrary
def lowerline 39library
def upperline 43library
def dominatingline 47library
def eligibleline 50library
def boundedline 53library
def stopline 56in the statementlibrary
def switchline 58library
def failureline 64library
def lowerFormline 69in the statementlibrary
def substituteline 73library
def cleanupline 76library
def advanceline 80library
def ForcedSingletonStepline 86in the statementlibrary
def initialline 93in the statementlibrary
def middleline 99in the statementlibrary
def finalStateline 105in the statementlibrary
def crossRowRestrictionline 110in the statementlibrary
def outputline 114in the statementlibrary
def betterline 116in the statementlibrary
theorem finite_execution_not_optimalline 118selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionnamespace TruffetV4Counterexampleabbrev Coeff := WithBot ℝabbrev Form := Fin 3 → Coeffabbrev Var := Fin 2abbrev RowIndex := Fin 2def column (j : Var) : Fin 3 := ⟨j.val, Nat.lt_trans j.isLt (by decide)⟩def nullForm : Form := fun _ => ⊥structure Row where  lhs : Form  rhs : Formstructure State where  rows : RowIndex → Row  cost : Form  remaining : Var → Propdef wellFormed (s : State) : Prop := ∀ j, ¬ s.remaining j →  s.cost (column j) = ⊥ ∧ ∀ i,    (s.rows i).lhs (column j) = ⊥ ∧ (s.rows i).rhs (column j) = ⊥def eval (f : Form) (w : Fin 3 → ℝ) : Coeff :=  max (f 0 + (w 0 : Coeff)) (max (f 1 + (w 1 : Coeff)) (f 2 + (w 2 : Coeff)))def feasible (s : State) (w : Fin 3 → ℝ) : Prop :=  ∀ i, eval (s.rows i).rhs w ≤ eval (s.rows i).lhs wdef lower (s : State) (i : RowIndex) (j : Var) : Prop :=  (s.rows i).lhs (column j) ≠ ⊥ ∧    (s.rows i).rhs (column j) < (s.rows i).lhs (column j)def upper (s : State) (i : RowIndex) (j : Var) : Prop :=  (s.rows i).rhs (column j) ≠ ⊥ ∧    (s.rows i).lhs (column j) < (s.rows i).rhs (column j)def dominating (s : State) (j : Var) : Prop :=  s.remaining j ∧ (∃ i, lower s i j) ∧ ∃ i, upper s i jdef eligible (s : State) (i : RowIndex) (j : Var) : Prop :=  s.remaining j ∧ lower s i j ∧ ((∃ k, dominating s k) → dominating s j)def bounded (s : State) : Prop :=  ∀ j, s.remaining j → (∃ i, lower s i j) ∨ ∃ i, upper s i jdef stop (s : State) : Prop := ∀ i, (s.rows i).rhs 2 ≤ (s.rows i).lhs 2def switch (s : State) : Prop :=  (∀ j, s.remaining j → s.cost (column j) = ⊥) ∧ ¬ stop s ∧  (∃ j, s.remaining j) ∧  (¬ ∃ i j, s.remaining j ∧ lower s i j) ∧  ∃ i j, s.remaining j ∧ upper s i jdef failure (s : State) : Prop := ¬ stop s ∧  ((¬ ∃ j, s.remaining j) ∨   ((¬ ∃ i j, s.remaining j ∧ lower s i j) ∧    (¬ ∃ i j, s.remaining j ∧ upper s i j)))def lowerForm (s : State) (i : RowIndex) (j : Var) : Form := fun k =>  if k = column j then ⊥ else    (s.rows i).rhs k + ((-WithBot.unbotD 0 ((s.rows i).lhs (column j)) : ℝ) : Coeff)def substitute (f : Form) (j : Var) (g : Form) : Form := fun k =>  if k = column j then ⊥ else max (f k) (f (column j) + g k)def cleanup (r : Row) : Row := by  classical  exact if ∀ k, r.rhs k ≤ r.lhs k then ⟨nullForm, nullForm⟩ else rdef advance (s : State) (i : RowIndex) (j : Var) : State where  rows := fun r => cleanup ⟨substitute (s.rows r).lhs j (lowerForm s i j),    substitute (s.rows r).rhs j (lowerForm s i j)⟩  cost := substitute s.cost j (lowerForm s i j)  remaining := fun k => s.remaining k ∧ k ≠ jdef ForcedSingletonStep (s s' : State) (i : RowIndex) (j : Var) : Prop :=  wellFormed s ∧ wellFormed s' ∧  bounded s ∧ ¬ stop s ∧ ¬ switch s ∧ ¬ failure s ∧  (∀ r k, eligible s r k ↔ r = i ∧ k = j) ∧  (∃ k, (substitute s.cost j (lowerForm s i j)) k ≠ ⊥) ∧  s' = advance s i jdef initial : State where  rows := ![⟨![(0 : Coeff), 0, ⊥], ![⊥, ⊥, (0 : Coeff)]⟩,    ⟨![(0 : Coeff), ⊥, ⊥], ![⊥, (1 : Coeff), ((-2 : ℝ) : Coeff)]⟩]  cost := ![(0 : Coeff), ⊥, ⊥]  remaining := fun _ => Truedef middle : State where  rows := ![⟨nullForm, nullForm⟩,    ⟨![(0 : Coeff), ⊥, ⊥], ![⊥, ⊥, (1 : Coeff)]⟩]  cost := ![(0 : Coeff), ⊥, ⊥]  remaining := fun j => j = 0def finalState : State where  rows := fun _ => ⟨nullForm, nullForm⟩  cost := ![⊥, ⊥, (1 : Coeff)]  remaining := fun _ => Falsedef crossRowRestriction (s : State) : Prop :=  ∀ i r : RowIndex, i ≠ r → ∀ α : ℝ, ∃ w : Fin 3 → ℝ,    eval (s.rows i).lhs w < (α : Coeff) + eval (s.rows r).rhs wdef output : Fin 3 → ℝ := ![1, 0, 0]def better : Fin 3 → ℝ := ![0, -1, 0]theorem finite_execution_not_optimal :    crossRowRestriction initial ∧    ForcedSingletonStep initial middle 0 1 ∧    ForcedSingletonStep middle finalState 1 0 ∧    (¬ ∃ j, finalState.remaining j) ∧ stop finalState ∧    feasible initial output ∧ feasible initial better ∧    output 2 = 0 ∧ better 2 = 0 ∧    (output 1 : Coeff) = eval (lowerForm initial 0 1) output ∧    (output 0 : Coeff) = eval (lowerForm middle 1 0) output ∧    eval initial.cost output = eval finalState.cost output ∧    eval initial.cost better < eval finalState.cost output := by  sorryend TruffetV4Counterexampleendend OAI

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.

Deterministic quasipolynomial-time mean-payoff games

Theorem

A uniform deterministic algorithm returns exactly the set

U=\bigl\{v\in V:\exists\text{ Max strategy }\sigma\ \forall\text{ Min strategies }\tau, \ \MP_w(\pi_{v,\sigma,\tau})\ge0\bigr\}

for every game specified above. For an absolute constant C>0, the complete computation uses at most

2^{C(\log_2(L+2))^2}

bit operations in a standard model polynomially equivalent to a uniform deterministic Turing machine. The bound includes preprocessing, exact arithmetic, recursion, and output of the entire set U.

Section 1, Introduction Read it in the paper TeX source, line 34

Show the TeX

A uniform deterministic algorithm returns exactly the set
\[
 U=\bigl\{v\in V:\exists\text{ Max strategy }\sigma\ 
       \forall\text{ Min strategies }\tau,
       \ \MP_w(\pi_{v,\sigma,\tau})\ge0\bigr\}
\]
for every game specified above. For an absolute constant $C>0$, the
complete computation uses at most
\[
                      2^{C(\log_2(L+2))^2}
\]
bit operations in a standard model polynomially equivalent to a
uniform deterministic Turing machine. The bound includes
preprocessing, exact arithmetic, recursion, and output of the entire
set $U$.

Randomized quasipolynomial-time mean-payoff games

Theorem

There is a uniform randomized algorithm that, for every such game, returns the entire set

U=\{v\in V:\text{there is a Max strategy }\sigma \text{ such that for every Min strategy }\tau,\ \operatorname{MP}_w(\pi_{v,\sigma,\tau})\ge0\}

correctly with probability at least 7/8. For an absolute constant C>0, on every random tape its total cost is at most

2^{C(\log_2(L+2))^2}

bit operations in a standard model polynomially equivalent to a uniform probabilistic Turing machine. The bound includes preprocessing, recursive calls, intermediate arithmetic, and random-bit generation.

Section 1, Introduction Read it in the paper TeX source, line 36

Show the TeX

There is a uniform randomized algorithm that, for every such game,
returns the entire set
\[
 U=\{v\in V:\text{there is a Max strategy }\sigma
       \text{ such that for every Min strategy }\tau,\ 
       \operatorname{MP}_w(\pi_{v,\sigma,\tau})\ge0\}
\]
correctly with probability at least $7/8$.
For an absolute constant $C>0$, on every random tape its total cost is at
most
\[
 2^{C(\log_2(L+2))^2}
\]
bit operations in a standard model polynomially equivalent to a uniform
probabilistic Turing machine. The bound includes preprocessing, recursive
calls, intermediate arithmetic, and random-bit generation.

Configuration and file facts

File: lean/ComparatorChallenges/TruffetCounterexample.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 104 link it.
Solution module: OAI.Optimization.Tropical.TruffetCounterexample
Allowed axioms: propext, Classical.choice, Quot.soundThe 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: 30 local vocabulary declarations
Lines of Lean, without comments or blank lines: 92
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