OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves \(e(R)\le e(S)\) for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity. This resolves Lech’s conjecture in every dimension and characteristic. (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.
The independent assessments find that the Lean statements express a supporting step or a different result.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 (R,\mathfrak m)\longrightarrow(S,\mathfrak n) be a flat local homomorphism of nonzero Noetherian local rings. Then
e(R)\leq e(S).
There is no restriction on the dimensions, the residue fields, or the characteristics of the rings.
No Lean statement in the release is paired with this theorem.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
We prove Lech’s multiplicity conjecture: Hilbert–Samuel multiplicity cannot decrease under a flat local homomorphism of nonzero Noetherian local rings. The result holds in arbitrary dimension, with no restrictions on the residue fields or characteristics.
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. The outside readers find that the release’s Lean states a supporting or different result, so the theorem it is paired with is shown under its own heading.
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.
DuttaDomain
The paper states
Corollary 5.4(Dutta multiplicity over positive-characteristic domains)
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.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
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 a supporting result (their term: “supporting-only”).
Their note:dutta_domain : DuttaDomainClaim is only a characteristic-p statement over a complete Noetherian local DOMAIN D (IsShortComplex D F →... Tendsto (duttaSequence D p F) atTop (𝓝 (duttaMultiplicity D p F)) ∧ multiplicity D ≤ duttaMultiplicity D p F); docs: ‘The paper’s full flat-local result in arbitrary characteristic is outside it’. The headline e(R) ≤ e(S) for flat local homomorphisms is not stated. multiplicity (Hilbert-Samuel via limUnder of d! * colength / N^d) is standard.
“supporting-only”, in their words: The challenge statements state a lemma or auxiliary result, not the headline.
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.
cert-machine finds that the Lean statements express a different result (their term: “DIFFERENT”).
The gap they name:The whole headline. The challenge is a characteristic-p lemma about one ring: for a complete Noetherian local domain D of characteristic p and a ‘short complex’ F (finite free, concentrated in degrees [−dim D, 0], finite-length homology, H⁰ ≠ 0), the Frobenius-twisted Euler characteristics p^{−n·dim D}·χ(F^n(F)) converge (the Dutta multiplicity) and e(D) is at most that limit. It is neither Lech’s inequality nor implied by it.
“DIFFERENT”, in their words: The challenge statements state a different theorem from the headline.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
Lech’s multiplicity conjecture compares Hilbert–Samuel multiplicities across flat local maps. The linked formalization covers a supporting characteristic-$p$ comparison over a complete Noetherian local domain $D$: for a complex satisfying the stated short-complex and finite-length homology conditions, its Frobenius multiplicity sequence converges to its Dutta multiplicity, and the Hilbert–Samuel multiplicity of $D$ is at most that Dutta multiplicity.
This selected statement is the complete-domain Dutta comparison. The paper’s full flat-local result in arbitrary characteristic is outside it.
AI-drafted comparisons, with no recorded human review
Unreviewed AI draft: states a supporting result. States a supporting characteristic-p comparison over complete local domains (Frobenius multiplicities converge to the Dutta multiplicity, which bounds the Hilbert–Samuel multiplicity). Lech’s inequality for flat local maps in arbitrary characteristic is 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.
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
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 “Career-defining publication”. The page says that nothing on it is verified.
From github.com/mahrud/openai-math-comments, 9 October 2026. Source
Points readers of the Lech family to a separate, human-written exposition of an AI-found proof of Lech’s conjecture in equal characteristic posted on 6 October.
Srikanth B. Iyengar, Linquan Ma, Mark Walker, Alex Gerko, Lech’s conjecture in equal characteristic: exposition of a proof found with ChatGPT Astra, 6 October 2026. Source
Independently of the release, a proof of Lech’s conjecture in equal characteristic found by Alex Gerko with ChatGPT Astra was written up by Iyengar, Ma and Walker, who say they believe the proofs correct but read them more as referees than as authors; the release’s family 194 claims the conjecture in full generality.
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 194The 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 194Atlas 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 1.1 (Lech’s conjecture)
In Lech’s multiplicity conjecture, 23 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
The challenge statements state a lemma or auxiliary result, not the headline.
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 a different theorem from the headline.
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.