Potential integral density on curve character varieties

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Resolves the determinant-one curve case of Litt’s integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SL_r character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes. (OpenAI’s Lean note)

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The paper states

Theorem 1.1

In Integral points on character varieties of curves, 25 September 2026

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 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.

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.
Papers and explanations: One current paper, readable here as a page beside its PDF.

Papers

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Integral points on character varieties of curves

25 September 2026, current edition.

Read it here PDF

We prove potential Zariski density of integral points on SL_r-character varieties of smooth complex curves in every rank. The result allows prescribed quasi-unipotent boundary monodromy, including exact nonsemisimple conjugacy classes, and holds on every component over the full ring of integers of one finite extension. Here integrality is measured in the ambient character variety, while the exact boundary conditions are imposed on the complex representation.

One Lean statement in this family

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`CharacterVarietiesAllSeamsSupport`

The paper states

Theorem 1.1

In Integral points on character varieties of curves, 25 September 2026

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

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.

The words they share: boundary, character, component, curve, dense, equation, extension, field, finite, number, prescribed, variety, zariski.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Compatibility of the produced marked solution with every seam equation

CharacterVarietiesAllSeamsSupport.lean, the release lists it as a main result.

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

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.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

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 a supporting result (their term: “supporting-only”).

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 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; the paper’s ‘normalized’ diagram; Diagram.Punctures R: a count and a unit of R per puncture per facet; 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 (, 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: 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 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 (, 264): equality of two coordinate frames as filtered spaces together with the identified graded quotients; strictly stronger than equality of flags

“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.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a supporting result (their term: “SUPPORT-ONLY”).

The gap they name: 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

“SUPPORT-ONLY”, in their words: The challenge statements state only supporting results.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The integral-density question asks whether integral points become Zariski dense in every component of a curve character variety after one finite extension of the number field. The linked formalization proves a supporting compatibility result for its marked-surface construction. Given a proper diagram with finitely many vertices, a chosen seam with maximal adjacent facet rank and a positive-genus parent, puncture data, an existing solution over a field, and a chosen boundary occurrence, the produced marked values satisfy every seam equation.
This is conditional compatibility of the produced solution. Potential integral density, including the prescribed-boundary and all-component conclusions, remains outside the selected statement.

AI-drafted comparisons, with no recorded human review

Compatibility of the produced marked solution with every seam equation CharacterVarietiesAllSeamsSupport
Unreviewed AI draft: states a supporting result. States a compatibility property of the paper’s marked-surface construction, one of the five supporting comparisons named in the Comparator README. Potential integral density itself is outside the statement.

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.

Compatibility of the produced marked solution with every seam equation `CharacterVarietiesAllSeamsSupport`

Two 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.	10 October 2026	Original report
MathVet	“comparator-sandboxed-pass”	Re-run through Comparator in a sandboxed virtual machine: accepted.	10 October 2026	Original report

What these papers cite in the release, and what cites them

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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

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22 references in Integral points on character varieties of curves

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What people have said

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 “A strong professor’s paper of the year”. The page says that nothing on it is verified.
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Identify the theorem the catalogue describes

Family 027 The 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

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