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Lean statement

Exact equality of deterministic and randomized logarithmic space

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.ExactDerandomization.exact_logarithmic_space_derandomization

Lean statement file: LogspaceEquality.leanShow on the mapFamily 103. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theoremWho ran the proof-verification programReport a comparison with the paper (GitHub issue)

Problems reported

  • Problem reported Reported as broken in the same way: its machine definitions are listed as holes, so a trivial replacement makes the checked statement vacuous. Alex Meiburg, Several of the comparator challenges in the OpenAI release are broken and hackable, 2026-10-08 Source

The Lean statement

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

theorem exact_logarithmic_space_derandomization :    L = RL ∧ RL = BPL := 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.ExactDerandomization.Word line 9

    abbrev Word := List Bool

  • abbreviation OAI.ExactDerandomization.Language line 10

    abbrev Language := Set Word

  • abbreviation OAI.ExactDerandomization.CoinTape line 11

    abbrev CoinTape := ℕ → Bool

  • inductive type OAI.ExactDerandomization.Direction line 13

    inductive Direction where  | left | stay | right  deriving DecidableEq

  • inductive type OAI.ExactDerandomization.InputSymbol line 17

    inductive InputSymbol where  | leftMarker | bit (b : Bool) | rightMarker  deriving DecidableEq

  • definition OAI.ExactDerandomization.Direction.move line 21

    def Direction.move (d : Direction) (z : ℤ) : ℤ :=  match d with  | .left => z - 1  | .stay => z  | .right => z + 1

  • definition OAI.ExactDerandomization.Direction.moveInput line 27

    def Direction.moveInput (d : Direction) {n : ℕ} (i : Fin (n + 2)) : Fin (n + 2) :=  match d with  | .left => ⟨i.val - 1, lt_of_le_of_lt (Nat.sub_le _ _) i.isLt⟩  | .stay => i  | .right => ⟨min (i.val + 1) (n + 1), by omega⟩

  • definition OAI.ExactDerandomization.readInput line 33

    def readInput (x : Word) (i : Fin (x.length + 2)) : InputSymbol :=  if i.val = 0 then .leftMarker else    match x[i.val - 1]? with    | some b => .bit b    | none => .rightMarker

  • structure OAI.ExactDerandomization.Action line 39

    structure Action (q w h : ℕ) where  nextState : Fin (q + 1)  write : Fin w → Bool  workMove : Fin w → Direction  inputMove : Fin h → Direction

  • structure OAI.ExactDerandomization.Machine line 45

    structure Machine (q w h : ℕ) where  initialState : Fin (q + 1)  output : Fin (q + 1) → Option Bool  transition : Fin (q + 1) → (Fin h → InputSymbol) → (Fin w → Bool) → Bool →    Action q w h

  • structure OAI.ExactDerandomization.Configuration line 51

    structure Configuration (q w h n : ℕ) where  state : Fin (q + 1)  inputPos : Fin h → Fin (n + 2)  workPos : Fin w → ℤ  work : Fin w → ℤ → Bool

  • definition OAI.ExactDerandomization.Machine.initial line 61

    def initial (M : Machine q w h) (n : ℕ) : Configuration q w h n where  state := M.initialState  inputPos := fun _ => ⟨0, by omega⟩  workPos := fun _ => 0  work := fun _ _ => false

  • definition OAI.ExactDerandomization.Machine.step line 67

    def step (M : Machine q w h) (x : Word) (b : Bool)    (c : Configuration q w h x.length) : Configuration q w h x.length :=
    Show the remaining 9 lines
      match M.output c.state with  | some _ => c  | none =>    let a := M.transition c.state (fun j => readInput x (c.inputPos j))      (fun k => c.work k (c.workPos k)) b    { state := a.nextState      inputPos := fun j => (a.inputMove j).moveInput (c.inputPos j)      workPos := fun k => (a.workMove k).move (c.workPos k)      work := fun k => Function.update (c.work k) (c.workPos k) (a.write k) }

  • definition OAI.ExactDerandomization.Machine.run line 79

    def run (M : Machine q w h) (x : Word) (coins : CoinTape) :    ℕ → Configuration q w h x.length  | 0 => M.initial x.length  | t + 1 => M.step x (coins t) (M.run x coins t)

  • definition OAI.ExactDerandomization.Machine.spaceThrough line 84

    def spaceThrough (M : Machine q w h) (x : Word) (coins : CoinTape) (t : ℕ) : ℕ :=  ∑ k : Fin w, ((Finset.range (t + 1)).image    (fun s => (M.run x coins s).workPos k)).card

  • definition OAI.ExactDerandomization.Machine.LogSpace line 88

    def LogSpace (M : Machine q w h) : Prop :=  ∃ c : ℕ, 0 < c ∧ ∀ (x : Word) (coins : CoinTape) (t : ℕ),    M.spaceThrough x coins t ≤ c * Nat.clog 2 (x.length + 2)

  • definition OAI.ExactDerandomization.Machine.Deterministic line 92

    def Deterministic (M : Machine q w h) : Prop :=  ∀ s i v, M.transition s i v false = M.transition s i v true

  • definition OAI.ExactDerandomization.Machine.HaltsBy line 95

    def HaltsBy (M : Machine q w h) (x : Word) (t : ℕ) : Prop :=  ∀ coins : CoinTape, ∃ b : Bool, M.output (M.run x coins t).state = some b

  • definition OAI.ExactDerandomization.Machine.Decides line 98

    def Decides (M : Machine q w h) (A : Language) : Prop :=  ∀ x : Word, ∃ (t : ℕ) (b : Bool),    (b = true ↔ x ∈ A) ∧ ∀ coins : CoinTape,      M.output (M.run x coins t).state = some b

  • definition OAI.ExactDerandomization.Machine.extendCoins line 103

    def extendCoins {t : ℕ} (bits : Fin t → Bool) : CoinTape :=  fun s => if hs : s < t then bits ⟨s, hs⟩ else false

  • definition OAI.ExactDerandomization.Machine.acceptanceProbability line 106

    def acceptanceProbability (M : Machine q w h) (x : Word) (t : ℕ) : ℚ :=  ((Finset.univ.filter (fun bits : Fin t → Bool =>    M.output (M.run x (extendCoins bits) t).state = some true)).card : ℚ) /    (2 : ℚ) ^ t

  • definition OAI.ExactDerandomization.polynomialClock line 113

    def polynomialClock (c k n : ℕ) : ℕ := c * (n + 2) ^ k

  • definition OAI.ExactDerandomization.L line 115

    def L : Set Language :=  {A | ∃ (q w h : ℕ) (M : Machine q w h),    M.Deterministic ∧ M.LogSpace ∧ M.Decides A}

  • definition OAI.ExactDerandomization.RL line 119

    def RL : Set Language :=  {A | ∃ (q w h : ℕ) (M : Machine q w h),    M.LogSpace ∧ ∃ (c k : ℕ), 0 < c ∧ ∀ x : Word,      M.HaltsBy x (polynomialClock c k x.length) ∧      (x ∈ A → (1 / 2 : ℚ) ≤ M.acceptanceProbability x        (polynomialClock c k x.length)) ∧      (x ∉ A → M.acceptanceProbability x (polynomialClock c k x.length) = 0)}

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

Definitions taken from OpenAI’s library 20

Comparator compares only the name and type of each definition listed here. When a theorem is configured, its proof uses the definition in OpenAI’s solution library. Read that definition as well as the one in this file when comparing the statement with the paper.

  • Comparator checks the name and type of Word; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of Language; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of CoinTape; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of move; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of moveInput; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of readInput; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of initial; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of step; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of run; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of spaceThrough; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of LogSpace; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of Deterministic; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of HaltsBy; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of Decides; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of extendCoins; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of acceptanceProbability; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of polynomialClock; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of L; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of RL; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.
  • Comparator checks the name and type of BPL; its definition is in OpenAI’s library, in the modules that OAI.Computability.Logspace.Equality imports.

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: acceptance, deterministic, machine, polynomial, probability, randomized, running, space and one more.

Theorem 1.2 of Exact derandomization of logarithmic space: L = RL = BPL

Theorem (Acceptance-probability approximation)

Fix a randomized machine \mathcal M with polynomial worst-case running time and O(\log(n+2)) work space. Let p_{\mathcal M}(x) be its acceptance probability on an input x of length n. There is a uniform deterministic algorithm which, on input (x,1^q) with q\ge1, outputs an integer 0\le a\le2^{q+2} satisfying

\left|a2^{-(q+2)}-p_{\mathcal M}(x)\right|\le2^{-q}.

It uses O(\log(n+2)+q) work space and (n+2)^{O(1)}2^{O(q)} time. The constants depend only on \mathcal M.

Section 1, Introduction Read it in the paper TeX source, line 40
Show the TeX
Fix a randomized machine $\mathcal M$ with polynomial worst-case
running time and $O(\log(n+2))$ work space. Let $p_{\mathcal M}(x)$ be
its acceptance probability on an input $x$ of length $n$. There is a
uniform deterministic algorithm which, on input $(x,1^q)$ with $q\ge1$,
outputs an integer $0\le a\le2^{q+2}$ satisfying
\[
       \left|a2^{-(q+2)}-p_{\mathcal M}(x)\right|\le2^{-q}.
\]
It uses $O(\log(n+2)+q)$ work space and
$(n+2)^{O(1)}2^{O(q)}$ time. The constants depend only on $\mathcal M$.

Other possible paper matches, closest wording first

  • Theorem 13.1 of Exact derandomization of logarithmic space: L = RL = BPL. Words of the Lean statement, its names split into words, that also appear in the theorem: 1/3, 2/3, decides, language, steps, tape and eleven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: explicit, randomized and bounds.
  • Corollary 12.4 of Exact derandomization of logarithmic space: L = RL = BPL. Words of the Lean statement, its names split into words, that also appear in the theorem: run, symbol, coin, halts, logarithmic, polynomial and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: time, randomized and bound.
  • Theorem 1.1 of Exact derandomization of logarithmic space: L = RL = BPL. Words of the Lean statement, its names split into words, that also appear in the theorem: derandomization, rl, bpl, logarithmic and space.
  • Theorem 12.1 of Exact derandomization of logarithmic space: L = RL = BPL. Words of the Lean statement, its names split into words, that also appear in the theorem: rational, polynomial, log, work, integer, machine and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: running, time and randomized.
  • Theorem 10.9 of Exact derandomization of logarithmic space: L = RL = BPL. Words of the Lean statement, its names split into words, that also appear in the theorem: rational, clock, range, bit, length, deterministic and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: numerical and bounded.
  • Corollary 12.3 of Exact derandomization of logarithmic space: L = RL = BPL. Words of the Lean statement, its names split into words, that also appear in the theorem: bpl.

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 10 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 348 s; peak memory 6.5 GiB).

    Reported 11 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.

Definitions compared by name and type only (20): Word, Language, CoinTape, move, moveInput, readInput, initial, step, run, spaceThrough, LogSpace, Deterministic, and 8 more. Their bodies come from OpenAI’s solution library.

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 103’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 disagree 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: Lean states L = RL ∧ RL = BPL for binary languages in an explicit multi-head, multi-tape logspace Turing-machine model (one fresh coin per step, polynomial clock on every coin tape, RL = 0 / >= 1/2, BPL = <= 1/3 / >= 2/3), which is the paper's Theorem (introduction.tex:28-32) after unfolding L, RL, BPL and chaining the two equalities; the binary-alphabet convention is the only routine step. The summary's second sentence (effective compiler with explicit polynomial running-time bounds, compiler.tex:8-19) has no Lean statement, as the scope note says.

    Definitions the group asks readers to compare with the paper: Word / Language: languages are sets of binary words (List Bool) only; the paper does not fix the alphabet in its Section 1, so the reduction from larger alphabets is a routine encoding step, not a restriction on the class. ; Machine / Action / Configuration: finite transition-table machine with q+1 states, w two-way-infinite binary work tapes (blank = false) and h read-only two-way input heads, with q, w, h chosen existentially; standard, and matches the paper's 'finite control, endmarked read-only input, finitely many work tapes' model (introduction.tex:10-15). ; Machine.step / run: one fresh coin bit per step (coins t used at step t+1), halting states are fixed points; matches 'fresh independent fair bits' and cannot reuse a past coin except via finite control. ; Machine.LogSpace: space = number of distinct cells visited by each work head up to time t, summed over tapes, bounded by c * Nat.clog 2 (|x|+2) for every input, every coin tape and every time t; standard counted-cell convention (blank-cell visits count), same as the paper. ; Machine.Decides: for every x some time t with all coin tapes halted with output b and (b = true <-> x in A); no explicit time bound is imposed on L-machines (polynomial time follows from the configuration count, as the paper notes at introduction.tex:20-22); standard. ; Machine.HaltsBy: halting by the clock on EVERY coin tape (worst-case time), as the paper requires ('All randomized running-time bounds hold on every random tape'); standard. ; Machine.acceptanceProbability: exact fraction of the 2^t coin strings of length t (padded with false) whose time-t state outputs true, with t = polynomialClock c k |x| = c*(|x|+2)^k; standard. ; RL / BPL: thresholds are exactly 0 on no and >= 1/2 on yes (RL), <= 1/3 on no and >= 2/3 on yes (BPL), as in introduction.tex:22-26; standard.

  • Carlos Toledo’s cert-machine

    Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.

    The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record

    Their note: The effective compiler and its explicit polynomial running-time bounds (headline's second sentence) are not stated; only the class equality is.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The selected theorem asserts equality of the language classes. The paper's explicit compiler and numerical running-time bounds are not separately stated in this Comparator target.”

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

142 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: 56 to its own declarations, 59 to Mathlib’s documentation. 25 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.

import Mathlibnamespace OAI/-! Exact logarithmic-space derandomization for finite transition-table machines. -/namespace ExactDerandomizationabbrev Word := List Boolabbrev Language := Set Wordabbrev CoinTape := ℕ → Boolinductive Direction where  | left | stay | right  deriving DecidableEqinductive InputSymbol where  | leftMarker | bit (b : Bool) | rightMarker  deriving DecidableEqdef Direction.move (d : Direction) (z : ℤ) : ℤ :=  match d with  | .left => z - 1  | .stay => z  | .right => z + 1def Direction.moveInput (d : Direction) {n : ℕ} (i : Fin (n + 2)) : Fin (n + 2) :=  match d with  | .left => ⟨i.val - 1, lt_of_le_of_lt (Nat.sub_le _ _) i.isLt⟩  | .stay => i  | .right => ⟨min (i.val + 1) (n + 1), by omega⟩def readInput (x : Word) (i : Fin (x.length + 2)) : InputSymbol :=  if i.val = 0 then .leftMarker else    match x[i.val - 1]? with    | some b => .bit b    | none => .rightMarkerstructure Action (q w h : ℕ) where  nextState : Fin (q + 1)  write : Fin w → Bool  workMove : Fin w → Direction  inputMove : Fin h → Directionstructure Machine (q w h : ℕ) where  initialState : Fin (q + 1)  output : Fin (q + 1) → Option Bool  transition : Fin (q + 1) → (Fin h → InputSymbol) → (Fin w → Bool) → Bool →    Action q w hstructure Configuration (q w h n : ℕ) where  state : Fin (q + 1)  inputPos : Fin h → Fin (n + 2)  workPos : Fin w → ℤ  work : Fin w → ℤ → Boolnamespace Machinevariable {q w h : ℕ}def initial (M : Machine q w h) (n : ℕ) : Configuration q w h n where  state := M.initialState  inputPos := fun _ => ⟨0, by omega⟩  workPos := fun _ => 0  work := fun _ _ => falsedef step (M : Machine q w h) (x : Word) (b : Bool)    (c : Configuration q w h x.length) : Configuration q w h x.length :=  match M.output c.state with  | some _ => c  | none =>    let a := M.transition c.state (fun j => readInput x (c.inputPos j))      (fun k => c.work k (c.workPos k)) b    { state := a.nextState      inputPos := fun j => (a.inputMove j).moveInput (c.inputPos j)      workPos := fun k => (a.workMove k).move (c.workPos k)      work := fun k => Function.update (c.work k) (c.workPos k) (a.write k) }def run (M : Machine q w h) (x : Word) (coins : CoinTape) :    ℕ → Configuration q w h x.length  | 0 => M.initial x.length  | t + 1 => M.step x (coins t) (M.run x coins t)def spaceThrough (M : Machine q w h) (x : Word) (coins : CoinTape) (t : ℕ) : ℕ :=  ∑ k : Fin w, ((Finset.range (t + 1)).image    (fun s => (M.run x coins s).workPos k)).carddef LogSpace (M : Machine q w h) : Prop :=  ∃ c : ℕ, 0 < c ∧ ∀ (x : Word) (coins : CoinTape) (t : ℕ),    M.spaceThrough x coins t ≤ c * Nat.clog 2 (x.length + 2)def Deterministic (M : Machine q w h) : Prop :=  ∀ s i v, M.transition s i v false = M.transition s i v truedef HaltsBy (M : Machine q w h) (x : Word) (t : ℕ) : Prop :=  ∀ coins : CoinTape, ∃ b : Bool, M.output (M.run x coins t).state = some bdef Decides (M : Machine q w h) (A : Language) : Prop :=  ∀ x : Word, ∃ (t : ℕ) (b : Bool),    (b = true ↔ x ∈ A) ∧ ∀ coins : CoinTape,      M.output (M.run x coins t).state = some bdef extendCoins {t : ℕ} (bits : Fin t → Bool) : CoinTape :=  fun s => if hs : s < t then bits ⟨s, hs⟩ else falsedef acceptanceProbability (M : Machine q w h) (x : Word) (t : ℕ) : ℚ :=  ((Finset.univ.filter (fun bits : Fin t → Bool =>    M.output (M.run x (extendCoins bits) t).state = some true)).card : ℚ) /    (2 : ℚ) ^ tend Machinedef polynomialClock (c k n : ℕ) : ℕ := c * (n + 2) ^ kdef L : Set Language :=  {A | ∃ (q w h : ℕ) (M : Machine q w h),    M.Deterministic ∧ M.LogSpace ∧ M.Decides A}def RL : Set Language :=  {A | ∃ (q w h : ℕ) (M : Machine q w h),    M.LogSpace ∧ ∃ (c k : ℕ), 0 < c ∧ ∀ x : Word,      M.HaltsBy x (polynomialClock c k x.length) ∧      (x ∈ A → (1 / 2 : ℚ) ≤ M.acceptanceProbability x        (polynomialClock c k x.length)) ∧      (x ∉ A → M.acceptanceProbability x (polynomialClock c k x.length) = 0)}def BPL : Set Language :=  {A | ∃ (q w h : ℕ) (M : Machine q w h),    M.LogSpace ∧ ∃ (c k : ℕ), 0 < c ∧ ∀ x : Word,      M.HaltsBy x (polynomialClock c k x.length) ∧      (x ∈ A → (2 / 3 : ℚ) ≤ M.acceptanceProbability x        (polynomialClock c k x.length)) ∧      (x ∉ A → M.acceptanceProbability x (polynomialClock c k x.length) ≤ (1 / 3 : ℚ))}theorem exact_logarithmic_space_derandomization :    L = RL ∧ RL = BPL := by  sorryend ExactDerandomizationend 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.

Exact derandomization of logarithmic space: L = RL = BPL

Theorem (Exact logarithmic-space derandomization)
\Lclass=\RL=\BPL.
Section 1, Introduction Read it in the paper TeX source, line 28
Show the TeX
\[
                 \Lclass=\RL=\BPL.
\]
Theorem (Acceptance-probability approximation)

Fix a randomized machine \mathcal M with polynomial worst-case running time and O(\log(n+2)) work space. Let p_{\mathcal M}(x) be its acceptance probability on an input x of length n. There is a uniform deterministic algorithm which, on input (x,1^q) with q\ge1, outputs an integer 0\le a\le2^{q+2} satisfying

\left|a2^{-(q+2)}-p_{\mathcal M}(x)\right|\le2^{-q}.

It uses O(\log(n+2)+q) work space and (n+2)^{O(1)}2^{O(q)} time. The constants depend only on \mathcal M.

Section 1, Introduction Read it in the paper TeX source, line 40
Show the TeX
Fix a randomized machine $\mathcal M$ with polynomial worst-case
running time and $O(\log(n+2))$ work space. Let $p_{\mathcal M}(x)$ be
its acceptance probability on an input $x$ of length $n$. There is a
uniform deterministic algorithm which, on input $(x,1^q)$ with $q\ge1$,
outputs an integer $0\le a\le2^{q+2}$ satisfying
\[
       \left|a2^{-(q+2)}-p_{\mathcal M}(x)\right|\le2^{-q}.
\]
It uses $O(\log(n+2)+q)$ work space and
$(n+2)^{O(1)}2^{O(q)}$ time. The constants depend only on $\mathcal M$.
Configuration and file facts
OpenAI’s Lean catalogue
OpenAI lists this as a main result in its Lean catalogue.
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
25 local vocabulary declarations
Lines of Lean, without comments or blank lines
100
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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Plectis Atlas

Exact equality of deterministic and randomized logarithmic space

Family 103: Exact derandomization of logarithmic space: \(\mathsf L=\mathsf{RL}=\mathsf{BPL}\)

Candidate pairing

Theorem (Acceptance-probability approximation)

Fix a randomized machine \mathcal M with polynomial worst-case running time and O(\log(n+2)) work space. Let p_{\mathcal M}(x) be its acceptance probability on an input x of length n. There is a uniform deterministic algorithm which, on input (x,1^q) with q\ge1, outputs an integer 0\le a\le2^{q+2} satisfying

\left|a2^{-(q+2)}-p_{\mathcal M}(x)\right|\le2^{-q}.

It uses O(\log(n+2)+q) work space and (n+2)^{O(1)}2^{O(q)} time. The constants depend only on \mathcal M.

Exact derandomization of logarithmic space: L = RL = BPL

TeX label thm:probability, build/sections/introduction.tex, line 40

The Lean statement says

theorem exact_logarithmic_space_derandomization :    L = RL ∧ RL = BPL := 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.

Comparator checks only the name and type of 20 definitions, among them Word; their bodies come from the solution.

Atlas selected this possible match because the two statements share words. No review of whether they mean the same thing is recorded.three independent groups report that Comparator accepted runs for this file at the release version shown here.Statement review: none recorded

25 local vocabulary declarationsAllowed axioms: propext, Quot.sound, Classical.choiceopenai/math at fd4aeebwcook04.github.io/plectis-atlas/statements/LogspaceEquality.html

Recorded comparisons with the paper

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Contributed Comparator run reports

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More about this statement

  • Commentary Aaronson counts L = BPL among the great derandomization conjectures short of P = BPP, whose truth was never seriously doubted but on which a whole subcommunity worked. Scott Aaronson, The Mathocalypse (Shtetl-Optimized), 2026-10-07 Source
  • Commentary Ran Raz (Princeton) says there were strong reasons to believe L = RL but they saw no promising direction before this proof. Scientific American (byline not recorded), The most exciting claims from OpenAI's heap of new proofs, 2026-10-08 Source

Questions to resolve when comparing this statement with the paper

  • 20 definition holes: Comparator compares these by name and type only, so a reader checks what each solution body means.
  • Two independent readers, MathVet and cert-machine, disagree on whether this family's challenges state its headline; a specialist reading would settle it.

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