OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves exponential lower bounds both for complementing two-way nondeterministic finite automata and for simulating one-way nondeterministic automata by two-way deterministic ones. The latter resolves the Sakoda–Sipser state-succinctness conjecture over growing finite alphabets; both results rule out polynomial state bounds independent of alphabet size. (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’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
For every integer n\ge4, set H=\{1,\ldots,n-2\} and \Sigma_n=\mathcal R_H. There is an n-state 2NFA A_n recognizing L_H such that every 2NFA recognizing \Sigma_n^*\setminus L(A_n) has at least
\frac12\,2^{\lfloor(n-4)/127\rfloor}-1
states. In particular, there is no polynomial state bound for complementation that is independent of the finite alphabet.
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
There are no absolute constants C,c>0 such that every n-state two-way nondeterministic automaton over every finite alphabet has an equivalent two-way deterministic automaton with at most Cn^c states.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
One more possible match
Theorem 1.1in An exponential two-way deterministic state lower bound for one-way liveness, 25 September 2026.
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.
We prove that two-way nondeterministic finite automata cannot be complemented with a polynomial number of states independent of the alphabet. For each n ≥ 4 we construct an n-state automaton over a finite alphabet whose complement requires at least \(\tfrac12 2^{\lfloor(n-4)/127\rfloor}-1\) states.
One-way liveness on h points accepts a word of binary relations when their ordered product is nonempty. For every h ≥ 2, it has a nondeterministic automaton with \(h+3\) states and no left moves, whereas every equivalent s-state two-way deterministic automaton satisfies \(4(s+2)^2\ge2^{\lfloor(h-2)/31\rfloor}\). Partial transition rules, stay moves, and nonaccepting infinite computations are allowed. The alphabets are finite and grow with h, so the result rules out an alphabet-independent polynomial state bound for deterministic two-way simulation, already for one-way nondeterministic sources.
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.
For every integer h\ge2, the language \OWL_h is recognized by a nondeterministic automaton with h+3 states using no left moves. Every equivalent deterministic two-way automaton with s states satisfies
4(s+2)^2\ge 2^{\lfloor(h-2)/31\rfloor}.
If an initially accepting configuration counts as acceptance, s+2 in (1.2) may be replaced by s+1.
The words they share: eq, configuration, owl, accepting, integer, recognized, one, acceptance, language, way, deterministic, nondeterministic, two, bound, states.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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”.
For every integer n\ge4, set H=\{1,\ldots,n-2\} and \Sigma_n=\mathcal R_H. There is an n-state 2NFA A_n recognizing L_H such that every 2NFA recognizing \Sigma_n^*\setminus L(A_n) has at least
\frac12\,2^{\lfloor(n-4)/127\rfloor}-1
states. In particular, there is no polynomial state bound for complementation that is independent of the finite alphabet.
The words they share: 1/2, complementation, integer, nfa, sigma, bound, finite, lower, two, least, state.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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”.
For every integer n\ge4, set H=\{1,\ldots,n-2\} and \Sigma_n=\mathcal R_H. There is an n-state 2NFA A_n recognizing L_H such that every 2NFA recognizing \Sigma_n^*\setminus L(A_n) has at least
\frac12\,2^{\lfloor(n-4)/127\rfloor}-1
states. In particular, there is no polynomial state bound for complementation that is independent of the finite alphabet.
The words they share: 1/2, integer, nfa, sigma, bound, state, finite, lower, two, complementation, least.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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”.
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:None; alphabet is SetRel (Fin (n-2)) (Fin (n-2)) (resp. BRel (Fin h)), so no alphabet-independent polynomial bound, exactly as the summary says.
Definitions the group asks readers to compare with the paper:Hand-rolled 2-way automata (TwoNFA, DMachine, NMachine) with endmarkers, stay moves and acceptance by an existential finite run (FiniteRun positive), a convention that differs from textbook halting-acceptance (documented in the docs and handled via the positive flag); ((n - 4) / 127) is natural-number floor division in the exponent
“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 paper asks how many states are needed to complement or determinize two-way nondeterministic finite automata. The formalization gives, for every $n\ge4$, an explicit $n$-state automaton whose complement requires at least $\tfrac12 2^{\lfloor(n-4)/127\rfloor}-1$ states. For the same family of relational languages and every $n\ge131$, every equivalent deterministic two-way automaton requires at least $\tfrac12 2^{\lfloor(n-4)/127\rfloor}$ states.
The alphabet grows with $n$, so neither complementation nor determinization has a polynomial state bound uniform over alphabets. The model has two endmarkers, left, right, and stay moves, and acceptance by a finite run. The separate results about one-way liveness and the Sakoda–Sipser problem are outside this scope.
The formalization gives a family of one-way nondeterministic automata with $h+3$ states whose languages require exponentially many states for two-way deterministic recognition. For every $h\ge2$, a deterministic recognizer with $s$ states satisfies $2^{\lfloor(h-2)/31\rfloor}\le4(s+2)^2$ under both acceptance conventions considered in the paper; one convention has the sharper factor $4(s+1)^2$. The alphabet may grow with $h$, so no alphabet-independent polynomial simulation bound holds.
AI-drafted comparisons, with no recorded human review
Exponential deterministic state lower bound for livenessOneWayLiveness
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the liveness paper: one-way nondeterministic automata with h + 3 states whose languages need exponentially many states for two-way deterministic recognition, with the explicit bound.
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem with its explicit bound: for every n ≥ 4, an n-state two-way nondeterministic automaton whose complement needs at least ½·2^{⌊(n−4)/127⌋} − 1 states.
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main complementation bound together with the paper’s determinisation corollary for the same witness, with the explicit constant ½ and threshold n ≥ 131 where the corollary states unspecified constants.
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.
Exponential deterministic state lower bound for livenessOneWayLiveness
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 proof and rates its significance, if it holds up, as “Major math journal”. 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
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 129The 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
8 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 76 s; peak memory 6.1 GiB).
Date
10 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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 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 52 s; peak memory 6.1 GiB).
Date
10 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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 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 54 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.
Exponential deterministic state lower bound for liveness
Role
the release lists it as a main result
Paired with
Theorem 1.1 (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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.