Weak normalization implies strong normalization in pure type systems
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a β-normal form, every β-reduction sequence terminates. This resolves the β-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts. (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.
Let \mathcal P=(\mathcal S,\mathcal A,\mathcal R) be any pure type system. If every legal expression of \mathcal P, in every valid context, is weakly \beta-normalizing, then every such expression is strongly \beta-normalizing. No functionality assumption is imposed on \mathcal A or \mathcal R.
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 prove that every weakly β-normalizing pure type system is strongly β-normalizing. Both properties quantify over all legal expressions in all valid contexts, and reduction acts inside type annotations. No functionality hypothesis is required. This resolves the β-Barendregt–Geuvers–Klop conjecture.
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.
Let \mathcal P=(\mathcal S,\mathcal A,\mathcal R) be any pure type system. If every legal expression of \mathcal P, in every valid context, is weakly \beta-normalizing, then every such expression is strongly \beta-normalizing. No functionality assumption is imposed on \mathcal A or \mathcal R.
The words they share: valid, strongly, beta, weakly, context, system, legal, normalizing, strong, weak, conjecture, barendregt, geuvers, klop, pure, expression.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
TypeSystemNormalization.lean, the release lists it as a supporting statement.
theoremweak_implies_strong {S : Type u} (P : Specification S) (h : SystemWeaklyNormalizing P) : SystemStronglyNormalizing P := 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:weak_implies_strong (P : Specification S) (h : SystemWeaklyNormalizing P) : SystemStronglyNormalizing P for every sort type S, arbitrary axioms/rule relations (nonfunctional allowed), annotated lam/pi syntax with β-reduction inside annotations (lam_domain, pi_domain, pi_body), typing rules ax/var/weaken/product/abstraction/application/conversion, valid contexts as lists, legal = ∃ A, M : A ∨ A : M, WN = some β-normal form, SN = Acc. Exactly the β-BGK conjecture including open contexts.
“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 proves the $\beta$-Barendregt–Geuvers–Klop conjecture for pure type systems: if every legal expression in every valid context has some terminating $\beta$-reduction sequence, then every $\beta$-reduction sequence from every such expression terminates. Reduction is allowed inside type annotations, and the specification may have arbitrary sorts and nonfunctional axioms or rules.
AI-drafted comparisons, with no recorded human review
Weak normalization implies strong normalization in every pure type systemTypeSystemNormalization
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: in every pure type system, weak β-normalisation of all legal expressions implies strong normalisation.
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.
Weak normalization implies strong normalization in every pure type systemTypeSystemNormalization
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.
Other people’s accounts of running proof-verification software, with links to their reports.
jfulfo, jfulfo/pts-normalization: Comparator run with nanoda for the pure type systems paper, 7 October 2026. Source
Repository carries a comparator-run.log for the nanoda-enabled challenge; outcome should be read from the log.
Talia Ringer, Jamie (student, surname not shown), pts-normalization: a confined copy of the OpenAI strong normalisation proof, 7 October 2026. Source
A type theory group reports that Comparator with the nanoda kernel accepts a confined copy of the normalisation proof, assuming the AI-assisted extraction of just that proof was done correctly.
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 “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
Fernando Chu, Comment on weak and strong normalisation in pure type systems, 7 October 2026. Source
A type theorist gives context that this was a major problem in pure type systems, says their own AI-assisted attempt had made no visible progress, and calls the result impressive.
Questions awaiting review
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 245The 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
Compare the proposed paper and Lean matches
Family 245Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1 (The \beta-Barendregt–Geuvers–Klop conjecture)
In Weak and strong normalization in pure type systems, 25 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
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 45 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.
Weak normalization implies strong normalization in every pure type system
Role
the release lists it as a supporting statement
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.