Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.
Theorem selected for verification:OAI.Lech.dutta_domain
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Definitions needed to read it
Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.
defDuttaDomainClaim : Prop := ∀ (D : Type u) [CommRing D] [IsNoetherianRing D] [IsLocalRing D] [IsDomain D] [IsAdicComplete (IsLocalRing.maximalIdeal D) D] (p : ℕ) [Fact p.Prime] [CharP D p] (F : CochainComplex (ModuleCat.{u} D) ℤ), IsShortComplex D F → (∀ n i,IsFiniteLength D ((frobeniusComplex D p n F).homology i)) ∧ Tendsto (duttaSequence D p F) atTop (𝓝 (duttaMultiplicity D p F)) ∧ multiplicity D ≤ duttaMultiplicity D p F
Possible paper match, with no recorded review
Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.
Words of the Lean statement, its names split into words, that also appear in the theorem: dutta, multiplicity, complete, short, complex, domain, local and noetherian. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: characteristic.
Let $(R,\mathfrak m)$ be a complete Noetherian local domain of
characteristic $p>0$, and let $F$ be a short complex over $R$.
Then
\[
\chi_\infty^R(F)\geq e(R).
\]
Other possible paper matches, closest wording first
Theorem 3.1 of Lech’s multiplicity conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: char, integer, extended, length, ideal, sequences and nine more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: characteristic.
Proposition 4.3 of Lech’s multiplicity conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: prime, dim, free, map, ideal, complete and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: flat and characteristic.
Theorem 2.1 of Lech’s multiplicity conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: length, free, sequences, modules, short, finite and two more.
Theorem 1.1 of Lech’s multiplicity conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: nonzero, dimensions, rings, local and noetherian. Found in the paper’s title, which its main theorem states: multiplicity. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: flat and characteristics.
Theorem 5.2 of Lech’s multiplicity conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: map, nonzero, rings, local and noetherian. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: flat and characteristic.
Theorem 7.4 of Lech’s multiplicity conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: nonzero, rings, zero, local and noetherian. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: flat and characteristic.
How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.
Reported Comparator runs
two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.
For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.
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 (.github/workflows/openai-math-kernel.yml), one j…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.
This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.
Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.
Do family 194’s Lean statements express the advertised result?
These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
Finds a supporting or a different result statedTheir term: “supporting-only”.
The challenge statements state a lemma or auxiliary result, not the headline. Their record
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.
Finds a supporting or a different result statedTheir term: “DIFFERENT”.
The challenge statements state a different theorem from the headline. Their record
Their note: 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.
OpenAI’s own note on what this Lean statement covers
“This selected statement is the complete-domain Dutta comparison. The paper's full flat-local result in arbitrary characteristic is outside it.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a supporting result”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
81 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.
Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.
Links in the file: 17 to its own declarations, 59 to Mathlib’s documentation. 7 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
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.
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.
Questions to resolve when comparing this statement with the paper
The AI draft finds no Lean statement in its family that states a paper's theorem as the paper does.