Separating choiceless counting from polynomial time and witnessed choice
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Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽3 defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot express. A separate result shows that adding witnessed symmetric choice strictly increases expressive power. Both separations hold for the full counting formalism, allowing hereditarily finite sets of arbitrary finite rank. (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.
Three other possible matches are listed below the card, among them Theorem 1.1 in Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting and Corollary 3 in the same paper.
The query Q on all finite structures in the fixed vocabulary \tau of eight binary relations is isomorphism-invariant and decidable in deterministic polynomial time, but is not definable in full choiceless polynomial time with counting.
Three more possible matches
Theorem 1.1in Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting, 24 September 2026.
Corollary 3in Choiceless polynomial time with counting does not capture polynomial time, 23 September 2026.
Proposition 2in Choiceless polynomial time with counting does not capture polynomial time, 23 September 2026.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We prove that choiceless polynomial time with counting does not capture polynomial time on unordered finite structures, confirming the noncapture conjecture of Blass, Gurevich and Shelah. A linear-consistency query over 𝔽3 in a fixed binary vocabulary is decidable in polynomial time but not in the full counting formalism.
We prove that witnessed symmetric choice strictly increases the expressive power of choiceless polynomial time with counting. A fixed sentence with one witnessed-choice occurrence defines a Boolean query on every finite input that is not definable in the original counting formalism.
Two Lean statements in this family
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The query Q on all finite structures in the fixed vocabulary \tau of eight binary relations is isomorphism-invariant and decidable in deterministic polynomial time, but is not definable in full choiceless polynomial time with counting.
The words they share: decidable, definable, query, time, full, polynomial, relations, finite, eight, choiceless, isomorphism, structures, invariant, counting.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
ChoicelessPolynomialTime.lean, the release lists it as a main result.
theoremmain : (∀ (A B : Type) [Fintype A] [Fintype B] (S : Input A) (T : Input B), S.Iso T → (S.query ↔ T.query)) ∧ OrdinaryPolynomialTime ∧ ¬ FullCPT.EvaluationDefinable (fun {_} [_] I => I.query) := 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”.
There exist a vocabulary \tau of eight binary relations, a fixed polynomial p, and a \CPT+\WSC sentence (\Phi,p) with one occurrence of a WSC operator such that:
(i)
(\Phi,p) has a Boolean value on every finite \tau-structure;
(ii)
no \CPT sentence over \tau has the same true-model class.
The \CPT sentences in (ii) may use counting and hereditarily finite sets of unrestricted finite rank.
The words they share: boolean, wsc, model, polynomial, relations, cpt, counting, sentences, finite, witnessed, choice, occurrence, one.
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:CPTSeparation.main : (∀ S T, S.Iso T → (S.query ↔ T.query)) ∧ OrdinaryPolynomialTime ∧ ¬ FullCPT.EvaluationDefinable (fun I => I.query), where query = consistency of an F₃ linear system coded by 8 relations, OrdinaryPolynomialTime uses Mathlib Turing.TM2ComputableInPolyTime with finite stack alphabets on ordered encodings, and EvaluationDefinable is polynomially bounded choiceless counting over hereditarily finite sets of arbitrary rank; and WitnessedChoice.main : ∃ φ, φ.formula.wscCount = 1 ∧ φ.BooleanOnAllInputs ∧ ∀ ψ, ψ.isCPT → φ.TrueModels ≠ ψ.TrueModels. Both separations of the summary are stated.
Definitions the group asks readers to compare with the paper:Custom ‘choiceless polynomial time with counting’, twice and in two different hand-written formalisms: FullCPT (ASM-style Program with Rule/Term over HF A = Lists A quotient, evaluationAccepts time space := ∃ h ≤ time(|A|),... (P.occurring rel h).card ≤ space(|A|)) in ChoicelessPolynomialTime, and the term/formula logic WitnessedChoice.BGS (Term.iterate, resource p n := ⌊p(n)⌋₊, Formula.wsc, isCPT φ := wscCount = 0) in WitnessedChoice; the nonexpressibility statements are only as good as these encodings of CPTC. ‘Polynomial time’ for the query itself is Mathlib’s standard TM2ComputableInPolyTime.
“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 formalized result separates polynomial time from choiceless polynomial time with counting. It gives an explicit query on finite structures with eight relations that is invariant under isomorphism and decidable in polynomial time, but is not definable in the stated hereditarily finite-set language with cardinality and polynomial bounds on stages and intermediate objects. The query applies to all finite inputs without a promise. The separate witnessed-symmetric-choice result is not included.
The formalization proves that witnessed symmetric choice strictly increases the expressive power of choiceless polynomial time with counting. It constructs one fixed sentence with exactly one witnessed-choice occurrence that gives a Boolean answer on every finite input, while no sentence of the original counting formalism defines the same class of inputs.
AI-drafted comparisons, with no recorded human review
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: an isomorphism-invariant polynomial-time query on eight-relation structures not definable in choiceless polynomial time with counting.
Strict expressive gain from one witnessed symmetric choiceWitnessedChoice
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: one sentence with a single witnessed symmetric choice defines a class that no sentence of choiceless polynomial time with counting defines.
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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 302 s; peak memory 6.3 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 184 s; peak memory 6.3 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.
Polynomial-time query outside choiceless polynomial time
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.
Strict expressive gain from one witnessed symmetric choice
Role
the release lists it as a supporting statement
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.