Compatibility of the produced marked solution with every seam equation (Lean)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem producedMarkedValues_allSeams :    D.ProducedMarkedSeamEquations q hproper hmax hg P x b0 i0 := by  sorry

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

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: boundary, character, component, curve, dense, equation, extension, field and five more.

Theorem 1.1 of Integral points on character varieties of curves

Theorem

Let X be a smooth connected complex algebraic curve and let r\geq1. For prescribed quasi-unipotent conjugacy classes C_1,\ldots,C_s as above, let K\subset\C be a number field containing their eigenvalues. There is a finite extension L/K such that

M_r(X)(\cO_L)\cap Y(\C)

is Zariski dense in Y, and hence in every irreducible component of Y. The same conclusion holds when only the characteristic polynomial of each boundary monodromy is prescribed, with all its roots roots of unity. It also holds for the entire character variety without boundary conditions; in that case one may take K=\Q.

For the relative statements and for projective X, one may take distinct primes p,q>r and

F=K(\sqrt2,\zeta_{pq}),\qquad L=F\bigl(\zeta_r,\{u^{1/r}:u\in\cO_F^\times\}\bigr),

where \zeta_m denotes a primitive mth root of unity. In the projective case there are no boundary conditions and K=\Q.

Section 1, Introduction Read it in the paper TeX source, line 35

Show the TeX

Let $X$ be a smooth connected complex algebraic curve and let $r\geq1$.
For prescribed quasi-unipotent conjugacy classes $C_1,\ldots,C_s$ as above,
let $K\subset\C$ be a number field containing their eigenvalues. There is
a finite extension $L/K$ such that
\[
 M_r(X)(\cO_L)\cap Y(\C)
\]
is Zariski dense in $Y$, and hence in every irreducible component of $Y$.
The same conclusion holds when only the characteristic polynomial of
each boundary monodromy is prescribed, with all its roots roots of unity.
It also holds for the entire character variety without boundary
conditions; in that case one may take $K=\Q$.

For the relative statements and for projective $X$, one may take distinct
primes $p,q>r$ and
\begin{equation}
 F=K(\sqrt2,\zeta_{pq}),\qquad
 L=F\bigl(\zeta_r,\{u^{1/r}:u\in\cO_F^\times\}\bigr),
\end{equation}
where $\zeta_m$ denotes a primitive $m$th root of unity. In the projective
case there are no boundary conditions and $K=\Q$.

Other possible paper matches, closest wording first

Theorem 4.1 of Integral points on character varieties of curves. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: solutions and surface.
Proposition 5.1 of Integral points on character varieties of curves. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: solutions and surface.

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.

Carlos Toledo’s cert-machine

How they ran it, in their words

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…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 10 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

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.

Allowed axioms: propext, Quot.sound, Classical.choice

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 027’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.

MathVet
Finds a supporting or a different result stated Their term: “supporting-only”.
The challenge statements state a lemma or auxiliary result, not the headline. Their record
Their note: The Lean states only a compatibility lemma inside the paper's cutting construction: for a proper diagram with finitely many vertices, a seam q whose parent facet has maximal rank and positive genus, punctures P and a K-valued solution x (K any field), the generator values produced for the diagram cut along a nonseparating band satisfy the framed-flag equation at every seam of the cut diagram. It says nothing about integral points, Zariski density, number fields or rings of integers, SL_r character varieties, or quasi-unipotent boundary classes, so it is a supporting step of the proof and not the headline.
Definitions the group asks readers to compare with the paper: Diagram F S V arity: solver-authored structure (Foundation/SurfacePresentation.lean:12-24) formalising the paper's surface diagram: PortAssembly (vertices of continuation/interchange/split-merge kinds wired to seams S with arities, facet colouring), rank, genus, boundary cycles; part of the statement because the challenge imports OAI ; Diagram.Proper: all facet ranks positive and every seam has at least two children (Foundation/BoundaryCycles.lean:252-254); the paper's 'normalized' diagram ; Diagram.Punctures R: a count and a unit of R per puncture per facet (SurfacePresentation.lean:62-64); quasi-unipotence is NOT imposed (docstring: imposed at the density stage), more general than the paper's roots-of-unity scalars ; Diagram.Solution P K: subtype of invertible-matrix assignments to generators satisfying the surface relations, the seam framed-flag equations and VertexHolds (SurfacePresentation.lean:108-112, 618-627); K is an arbitrary field with Algebra R K, so this is a field-valued point, not an integral point ; [Finite V]: only the vertex type is assumed finite; seams are in bijection with port endpoints (PortAssembly.attach) hence finite, facets F are not assumed finite ; hmax and hg: the parent facet of seam q has maximal rank among all facets, and positive genus ; ProducedMarkedSeamEquations (Seams/ProducedEquations.lean:25-30): for every seam s of the produced marked (cut) diagram C, SameFramedFlag (C.seamGrade s) of matrixUnitEquiv of the left and right seam words evaluated at the produced values; a Prop-valued def built from solver-authored producedMarkedDiagram and producedMarkedValues ; cutGeneratorWitness (Frames/MarkedFrames.lean:18-24) is declared opaque (body: identifiedBand ...), so the produced diagram and values depend on this chosen simultaneous splitting and cannot be unfolded by rfl; not an axiom ; SameFramedFlag / matrixUnitEquiv (Foundation/Flags.lean:155, 264): equality of two coordinate frames as filtered spaces together with the identified graded quotients; strictly stronger than equality of flags
Carlos Toledo’s cert-machine
Finds a supporting or a different result stated Their term: “SUPPORT-ONLY”.
The challenge statements state only supporting results. Their record
Their note: the entire headline: potential Zariski density of integral points on SL_r character varieties is not stated; the challenge proves only that the produced marked values satisfy every seam equation of the produced marked diagram

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The linked formalization proves a supporting compatibility result for its marked-surface construction.”
Quoted from OpenAI’s Lean note for this family

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

25 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, 7 to Mathlib’s documentation. 1 more goes to the OpenAI modules the file imports. 1 declared name also links 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.

import OAI.AlgebraicGeometry.CharacterVarieties.Seams.ProducedEquationsnamespace OAInoncomputable sectionnamespace IntegralCharacterVarieties.SurfacePresentation.Diagramopen scoped Classical Matrixopen OccurrenceIncidence VertexTable MatrixExpression NamedBandGrades HomTransportvariable {F S V R K : Type} {arity : S → ℕ} [CommRing R] [Field K] [Algebra R K]    (D : Diagram F S V arity) (q : S) [Finite V]    (hproper : D.Proper) (hmax : ∀ f, D.rank f ≤ D.rank (D.ports.facet ⟨q,none⟩))    (hg : 0 < D.genus (D.bandCutParent q))    (P : D.Punctures R) (x : D.Solution P K)    (b0 : Fin (D.boundaryCount (D.bandCutParent q)))    (i0 : Fin (D.boundaryLength (D.bandCutParent q) b0))theorem producedMarkedValues_allSeams :    D.ProducedMarkedSeamEquations q hproper hmax hg P x b0 i0 := by  sorryend IntegralCharacterVarieties.SurfacePresentation.Diagramendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Integral points on character varieties of curves