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.
Theorems selected for verification:OAI.SiegelZeros.WeightedTorusJets.exists_absolute_real_zero_gapOAI.SiegelZeros.WeightedTorusJets.dirichletRealZeroBound_proof
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.
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: log, dirichlet, absolute, primitive, real, zero and function. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonprincipal, conductor, beta and constant.
Statements in Uniform exclusion of Landau–Siegel zeros come first because the release’s name for this Lean statement shares words with that paper’s title.
Theorem 1 of Uniform exclusion of Landau–Siegel zeros
Theorem
There is an absolute constant c>0 such that every real zero \beta\in(0,1) of every primitive nonprincipal real Dirichlet L-function of conductor q\ge3 satisfies
There is an absolute constant $c>0$ such that every real zero
$\beta\in(0,1)$ of every primitive nonprincipal real Dirichlet
$L$-function of conductor $q\ge3$ satisfies
\begin{equation}
(1-\beta)\log q\ge c.
\end{equation}
Other possible paper matches, closest wording first
Proposition 16.1 of The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane \(\Re s\gt 7/8\). Words of the Lean statement, its names split into words, that also appear in the theorem: im, proof, primitive, character and zero. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonprincipal and conductor.
Theorem 1.1 of The Quasi-Riemann Hypothesis (alternate 11/12 proof). Words of the Lean statement, its names split into words, that also appear in the theorem: dirichlet, zeros and function. Found in the paper’s title, which its main theorem states: proof.
Theorem 1.1 of The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane \(\Re s\gt 7/8\). Words of the Lean statement, its names split into words, that also appear in the theorem: dirichlet, character, zero and function.
Proposition 11.3 of The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane \(\Re s\gt 7/8\). Words of the Lean statement, its names split into words, that also appear in the theorem: dirichlet, primitive, zero and function. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
Corollary 1.2 of The Quasi-Riemann Hypothesis (alternate 11/12 proof). Words of the Lean statement, its names split into words, that also appear in the theorem: log, absolute and function. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constant.
Proposition 15.2 of The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane \(\Re s\gt 7/8\). Words of the Lean statement, its names split into words, that also appear in the theorem: weight, bound and real. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
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.
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 003’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.
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: No gap for zeta/Dirichlet L (Mathlib objects). Hecke 7/8 clause rests on a custom WeakFEPair-defined L-function (LFunction := continuedLattice χ s / 6) whose identification with the Dirichlet series is not in the statement; 11/12 companion not stated separately (implied).
Definitions the group asks readers to compare with the paper: Hecke part: def LFunction (χ : Character) (s : ℂ) : ℂ := continuedLattice χ s / 6 with continuedLattice χ = HeckeTheta.latticeL (coefficients χ) (custom theta/WeakFEPair construction; custom structure Character with modulus, residue : MulChar (O ⧸ modulus) ℂ, unit_trivial, period); no lemma linking it to Σ χ(a) N(a)^{-s}
OpenAI’s own note on what this Lean statement covers
“There is one constant $c>0$ such that every primitive nonprincipal real Dirichlet character of conductor $q\ge3$ and every real zero $0<\beta<1$ of its $L$-function satisfy $1-\beta\ge c/\log q$.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
30 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: 0 to its own declarations, 9 to Mathlib’s documentation. 2 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
Every finite-order Hecke L-function over F=\mathbb Q(\sqrt{-3}) has no zero in \Re s>7/8. The same holds for every Dirichlet L-function, including \zeta(s). A pole at s=1 for a principal character is allowed.
Every finite-order Hecke $L$-function over
$F=\mathbb Q(\sqrt{-3})$ has no zero in $\Re s>7/8$.
The same holds for every Dirichlet $L$-function, including $\zeta(s)$.
A pole at $s=1$ for a principal character is allowed.
Corollary
There are absolute constants A,C>0 such that, for every odd prime p, the least positive quadratic nonresidue n(p) satisfies
n(p)\le C(\log p)^A.
In particular, n(p)\ll_\delta p^\delta for every \delta>0, proving Vinogradov’s least quadratic nonresidue conjecture [TaoNonresidues, Conjecture 1.1]. Given an odd prime p and a\in\mathbb F_p in binary representation, there is a deterministic algorithm, with running time polynomial in \log p, that returns a square root of a or reports that none exists.
There are absolute constants $A,C>0$ such that, for every odd prime $p$,
the least positive quadratic nonresidue $n(p)$ satisfies
\[
n(p)\le C(\log p)^A.
\]
In particular, $n(p)\ll_\delta p^\delta$ for every $\delta>0$, proving
Vinogradov's least quadratic nonresidue conjecture
\cite[Conjecture 1.1]{TaoNonresidues}.
Given an odd prime $p$ and $a\in\mathbb F_p$ in binary representation,
there is a deterministic algorithm, with running time polynomial in
$\log p$, that returns a square root of $a$ or reports that none exists.
There is an absolute constant c>0 such that every real zero \beta\in(0,1) of every primitive nonprincipal real Dirichlet L-function of conductor q\ge3 satisfies
There is an absolute constant $c>0$ such that every real zero
$\beta\in(0,1)$ of every primitive nonprincipal real Dirichlet
$L$-function of conductor $q\ge3$ satisfies
\begin{equation}
(1-\beta)\log q\ge c.
\end{equation}
propext, Classical.choice, Quot.soundThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports
Mathlib
Imports from outside Mathlib
None
Local vocabulary
Imports only Mathlib; no local vocabulary added
Lines of Lean, without comments or blank lines
16
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Uniform real-zero gap
Family 003: The quasi-Riemann hypothesis
Candidate pairing
Theorem
There is an absolute constant c>0 such that every real zero \beta\in(0,1) of every primitive nonprincipal real Dirichlet L-function of conductor q\ge3 satisfies
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.
More about this statement
Commentary Hector Pasten (PUC Chile) says the quasi-Riemann hypothesis is enough for many applications of the full hypothesis and that the Siegel-zero result settles that question for zeta and Dirichlet L-functions. Scientific American (byline not recorded), The most exciting claims from OpenAI's heap of new proofs, 2026-10-08Source