One-tape time simulation in two-fifths-power space
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time T using \(O(T^{2/5}\log^C(T+2))\) space, improving the square-root exponent. Heads move at most one cell per step; finitely many read-only input heads are allowed. Initial contents are independent of T, and contents and input symbols have polylogarithmic-space access. Simulation time is unrestricted. (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
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.
Fix a deterministic machine with one writable tape and head, a fixed number of read-only input heads, unit movement and fixed origins for all heads, cap-independent symbol functions, and a fixed accessor as above. For every public input x and supplied binary cap T\ge2 for which the access condition holds, its finite-control and halting outcome by time T can be computed deterministically in \softO(T^{2/5}) work-space bits, excluding storage for the read-only input. Simulation time is unrestricted, and the final writable tape need not be materialized. If the access condition holds for (x,T) at every T\ge2, and the machine halts on x after t steps without a supplied running time, its outcome can be computed in \softO((t+2)^{2/5}) work-space bits.
Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
We show that a fixed deterministic Turing machine with one writable tape and head can be simulated in \(O(T^{2/5}\mathop{\mathrm{polylog}}\nolimits (T+2))\) work-space bits when a binary time cap T ≥ 2 is supplied. The simulator computes the finite-control and halting outcome by time T; its running time is unrestricted. The result allows a fixed number of read-only input heads and requires a fixed accessor that supplies every initial writable and read-only symbol within distance T of the relevant head origin in polylogarithmic space. This improves the square-root space exponent for one-tape machines, answering Williams’s question for this model.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
Fix a deterministic machine with one writable tape and head, a fixed number of read-only input heads, unit movement and fixed origins for all heads, cap-independent symbol functions, and a fixed accessor as above. For every public input x and supplied binary cap T\ge2 for which the access condition holds, its finite-control and halting outcome by time T can be computed deterministically in \softO(T^{2/5}) work-space bits, excluding storage for the read-only input. Simulation time is unrestricted, and the final writable tape need not be materialized. If the access condition holds for (x,T) at every T\ge2, and the machine halts on x after t steps without a supplied running time, its outcome can be computed in \softO((t+2)^{2/5}) work-space bits.
The words they share: 2/5, steps, functions, cap, finite, public, access, tape, binary, outcome, head, read, symbol, unit, work, bits, halts, input, time, simulation, independent, control, deterministic, machine, writable, one, space.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
OneTapeSpace.lean, the release lists it as a main result.
theoremcapped [Fintype X] (A : M.Sym) (p q : ℝ) (hp : 1 ≤ p) (_hq : 1 ≤ q) : ∃ (N : TM X Unit M.Outcome) (e : ℕ) (c : ℝ), 0 < c ∧ ∀ (x : List X) (T : ℕ), 2 ≤ T → M.Access A p q x T → ∃ b : ℕ, N.Within (binaryIn x (fun _ => T)) (M.outcome x T) b ∧ (b:ℝ) ≤ c*(T:ℝ)^exponent*(Log T)^e := bysorry
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”.
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:Lean capped and unknown are the two clauses of the paper’s single main theorem with Single = the fixed one-writable-tape machine, Access = the polylog-space accessor condition, outcome = (finite-control state, halted flag) with absorbing halting states, Within = work-space bits with no time bound, and c*T^(2/5)*(log2(T+2))^e = O~(T^{2/5}); no step beyond unfolding these definitions is needed. The summary’s remark ‘improving the square-root exponent’ is a comparison with the classical bound, not a separate formal claim.
Definitions the group asks readers to compare with the paper:Single.sensor: initial writable-tape contents (tag none) and the symbols read by each of the heads read-only heads (tag some i) are an arbitrary function List X -> Option (Fin heads) -> Z -> A, constrained only through the Access hypothesis; input symbols share the tape alphabet A. This is the paper’s ‘uniform interface’, not a restriction. The tape contents sensor x none do not depend on T by construction.; Single.Access: the polylog-space accessor hypothesis is part of the statement (the theorem is conditional on it, per pair (x,T), as in the paper): a fixed uniform TM A, given x, the cap T, the tag, and the sign and magnitude of l in binary, outputs sensor x h l within p*(log2(T+2))^q work bits for every tag and every |l| <= T. In unknown it is required for every T >= 2.; TM / TM.Within / Window.bits: simulator and accessor model is a multi-tape machine (finite Q, finitely many work tapes with finite alphabets) with endmarked read-only heads on the public word and on each binary argument tape; space = sum over work tapes of (window span including the origin) * bit-width of the alphabet; input and cap tapes are not charged; NO time bound (matches ‘simulation time is unrestricted’); standard counted-cell convention.; Single.move / Single.outcome: halted states are absorbing (state, tape symbol and head positions unchanged), and outcome x n = (state, halted flag) after exactly n steps, so it equals the ‘finite-control and halting outcome by time n’.; [Fintype X]: the public alphabet is finite.; Cost.Log / Cost.exponent: Log T = Real.log (T+2) / Real.log 2 (binary log of T+2); exponent = 2/5 used as a real power; the log exponent e is a natural number and c a positive real, both existentially quantified before x and T (they may depend on M, A, p, q only), matching O~ with ‘a fixed constant C’.; Mark.see / Mark.walk: read-only inputs are endmarked (0 = left marker, 1..len = symbols, len+1 = right marker) and heads are confined to [0, len+1]; standard.
“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.
The formalization gives two-fifths-power space simulation of a fixed deterministic machine with one writable tape and finitely many read-only input heads. Heads move at most one cell per step, initial contents are independent of the time bound, and contents and input symbols must be accessible in polylogarithmic space. Given a binary time cap $T\ge2$, the simulator returns the source’s finite-control outcome and halting flag using $O(T^{2/5}(\log(T+2))^C)$ space.
A second simulator needs no time cap: on an input that halts at time $t$, it returns the halted outcome using $O((t+2)^{2/5}(\log(t+2))^C)$ space. Simulation time is unrestricted. General multitape simulation, hierarchy consequences, and bounded-space-halting results are outside these selected statements.
AI-drafted comparisons, with no recorded human review
Time-capped and unknown-time one-tape simulation in two-fifths-power spaceOneTapeSpace
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a one-tape machine running for time T is simulated in O(T^{2/5} polylog T) space, in a time-capped form and a form without a cap. Multitape and hierarchy consequences are outside it.
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.
Time-capped and unknown-time one-tape simulation in two-fifths-power spaceOneTapeSpace
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original 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.
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.
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.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a algorithm and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 137The 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
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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
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; 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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
10 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 91 s; peak memory 6.2 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
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.
Time-capped and unknown-time one-tape simulation in two-fifths-power space
Role
the release lists it as a main result
Paired with
Theorem 1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.