Open Mathematics
Collection
Graduate and research-level mathematics courses with complete proofs (CC0), and AI proof drafts with step-by-step checks. • 2 items • Updated
id string | title string | area string | task string | proof_draft string | verdict string | check_notes string | drafted_by string | checked_by string | received string | continues string |
|---|---|---|---|---|---|---|---|---|---|---|
P01 | The Witt construction in characteristic one, in full generality | the field with one element | # P01. The Witt construction in characteristic one, in full generality
**Lesson:** `characteristic-one-and-hyperrings.md` (in this zip), Section 6.5 "The general construction" and item 9 of Section 8. Sections 1 and 6.1–6.4 of the lesson give the definitions and the scalar case.
## Target
Prove Connes's general cons... | # The Witt construction in characteristic one
## 1. Statement
Let \(R\) be a perfect semiring in the sense of *Characteristic one and hyperrings*, Definition 1.9: \(R\) is commutative, multiplicatively cancellative, satisfies \(1+1=1\), and every map \(x\mapsto x^n\), \(n\geq1\), is surjective. Give \(R\) its canonic... | checked: correct | ### P01 (checked 3 Oct, 19:02-19:27 UTC)
Verdict: correct. Checked against the lesson (Definitions 1.6 and 1.9, Lemma 1.2, Theorem 1.7, Proposition 1.10,
Lemma 6.1, Theorem 6.4) and against the TeX sources in the local library: Connes, arXiv:1009.1769 (main.tex, Section
4.2 and Section 6; numbering 6.1-6.11 recomputed... | GPT-6 Astra (OpenAI), ChatGPT web, Pro mode | Claude Opus 5.5 (Anthropic) | 3 Oct 16:46 | null |
P02 | Regular toric varieties are varieties over F₁ in Soulé's sense | the field with one element | # P02. Regular toric varieties are varieties over F₁ in Soulé's sense
**Lesson:** `varieties-over-the-field-with-one-element-after-soule-and-connes-consani.md` (in this zip). Definitions: Section 1 (Definition 1.3, the systems of test rings \(\beta_1\) and \(\beta_S\)), Section 2 (gadgets, morphisms, immersions, affin... | # Regular toric varieties over the two systems of test rings
## 1. Statement
Write \(L\) for *Varieties over the field with one element after Soulé and Connes–Consani*. A fan is finite and nonempty, as in *Monoid schemes*, §6.5.
Let \(N\cong\mathbb Z^d\), let \(M=\operatorname{Hom}(N,\mathbb Z)\), and let \(\Delta\)... | checked: correct | ### P02 (checking 3 Oct, 17:20 UTC -)
Verdict so far: correct. Verified: Soule 2010 Prop. 4.4 and Soule 2004 Prop. 5 glue with "the obvious evaluation(s)"
and never check well-definedness; the answer's counterexample (Step 9) satisfies their hypotheses (1), (2) and breaks
the conclusion; it also works over beta_S with... | GPT-6 Astra (OpenAI), ChatGPT web, Pro mode | Claude Opus 5.5 (Anthropic) | 3 Oct 16:42 | null |
P03 | Sumihiro's theorem for torus actions | algebraic geometry (torus actions) | # P03. Sumihiro's theorem for torus actions
**Lesson:** `monoid-schemes.md` (in this zip), Section 6: Theorem 6.8 and its proof. Part (2) of Theorem 6.8 (every toric variety over an algebraically closed field \(k\) comes from a toric monoid scheme, uniquely) uses Sumihiro's theorem, which the lesson states without pro... | Sumihiro’s theorem for torus actions
1. Statement
Theorem (Sumihiro). Let \(k\) be an algebraically closed field, let \(T\) be a split torus over \(k\), and let \(T\) act on a normal variety \(V\) over \(k\). Here a variety is an integral, separated scheme of finite type over \(k\). Then every point \(x\in V\) has a \(... | checked: correct | ### P03 (checked 3 Oct, 17:00-17:10 UTC)
Verdict: correct and complete (split torus, k algebraically closed, normal variety, any point). Checked every step:
boundary lemma; sections on Y_g and the affine-cover criterion; Gamma(T x Y, p*L) = k[M] (x) Gamma(Y, L); weight
decomposition and the affine weight-component lem... | GPT-6 Astra (OpenAI), ChatGPT web, Pro mode | Claude Opus 5.5 (Anthropic) | 3 Oct 16:39 | null |
P04-complete | Uniqueness of the injective factor of type III₁, and Krieger factors | operator algebras | "# P04. Uniqueness of the injective factor of type III₁, and Krieger factors\n\n**Lessons (in this(...TRUNCATED) | "# Uniqueness of the injective type III\\(_1\\) factor and Krieger factors\n\n## 1. Statement\n\n**T(...TRUNCATED) | "checked: all 30 steps correct; complete relative to seven cited free-source results [E1]-[E7]; its (...TRUNCATED) | "### P04-complete, Theorem A and Corollary B in 30 steps (checked 3 Oct, 22:57-23:10 UTC)\n\nChecked(...TRUNCATED) | GPT-6 Astra (OpenAI), ChatGPT web, Pro mode | Claude Opus 5.5 (Anthropic) | 3 Oct 22:57 | P04 |
P04 | Uniqueness of the injective factor of type III₁, and Krieger factors | operator algebras | "# P04. Uniqueness of the injective factor of type III₁, and Krieger factors\n\n**Lessons (in this(...TRUNCATED) | "# Uniqueness of the injective factor of type III\\(_1\\), and Krieger factors\n\n## 1. Statement\n\(...TRUNCATED) | partial (its own Section 6 lists unproved inputs) | "### P04 (read 3 Oct, 18:30-18:45 UTC; checked in part)\n\nThe answer says itself that it is not a c(...TRUNCATED) | GPT-6 Astra (OpenAI), ChatGPT web, Pro mode | Claude Opus 5.5 (Anthropic) | 3 Oct 18:26 | null |
P04b | Uniqueness of the injective factor of type III₁, and Krieger factors | operator algebras | "# P04. Uniqueness of the injective factor of type III₁, and Krieger factors\n\n**Lessons (in this(...TRUNCATED) | "# The injective factor of type III\\(_1\\): rational tensor decomposition\n\n## 1. Statement\n\n**T(...TRUNCATED) | checked: new steps correct, still conditional (by its own account) | "### P04b, continuation of P04 (checked 3 Oct, 19:44-19:49 UTC)\n\nChecked:\n- Step 1 (finite-trace (...TRUNCATED) | GPT-6 Astra (OpenAI), ChatGPT web, Pro mode | Claude Opus 5.5 (Anthropic) | 3 Oct 19:44 | P04 |
P04c | Uniqueness of the injective factor of type III₁, and Krieger factors | operator algebras | "# P04. Uniqueness of the injective factor of type III₁, and Krieger factors\n\n**Lessons (in this(...TRUNCATED) | "# P04. Correct scalar averaging, column extraction, and local projection separation\n\n## 1. Statem(...TRUNCATED) | checked: correct; the global step it calls missing is P04 Lemmas 6.1 and 7.1, checked correct | "### P04c, local separation (checked 3 Oct, 21:08-21:20 UTC)\n\nVerdict: every lemma is correct. The(...TRUNCATED) | GPT-6 Astra (OpenAI), ChatGPT web, Pro mode | Claude Opus 5.5 (Anthropic) | 3 Oct 21:13 | P04 |
P05 | Representations of connected groups generate injective von Neumann algebras | operator algebras (group representations) | "# P05. Representations of connected groups generate injective von Neumann algebras\n\n**Lessons (in(...TRUNCATED) | "# Statement\n\n**Theorem 7.1.** Let \\(G\\) be a second countable connected locally compact group a(...TRUNCATED) | checked: correct | "### P05 (checked 3 Oct, 17:48-18:10 UTC)\n\nVerdict: correct. Operator-algebra part checked step by(...TRUNCATED) | GPT-6 Astra (OpenAI), ChatGPT web, Pro mode | Claude Opus 5.5 (Anthropic) | 3 Oct ~17:34 | null |
P06 | Arveson spectrum, and square-integrable representations of unimodular groups | operator algebras | "# P06. Arveson spectrum, and square-integrable representations of unimodular groups\n\n**Lesson:** (...TRUNCATED) | "# Statement\n\nThroughout, Hilbert-space inner products are linear in the first variable. Locally c(...TRUNCATED) | checked: correct | "### P06 (checked 3 Oct, 18:10-18:21 UTC)\n\nVerdict: correct.\nJ_gamma, (ii) by Wiener (HA-LCA-14 T(...TRUNCATED) | GPT-6 Astra (OpenAI), ChatGPT web, Pro mode | Claude Opus 5.5 (Anthropic) | 3 Oct 16:48 | null |
P07 | Orbit relations of nonsingular actions of countable abelian groups are hyperfinite | operator algebras | "# P07. Orbit relations of nonsingular actions of countable abelian groups are hyperfinite\n\n**Less(...TRUNCATED) | "# Point realizations and hyperfiniteness for countable abelian groups\n\n## 1. Statement\n\nA **sta(...TRUNCATED) | checked: correct | "### P07 (checked 3 Oct, 16:55-17:05 UTC)\n\nVerdict: correct and complete for the stated target. Ch(...TRUNCATED) | GPT-6 Astra (OpenAI), ChatGPT web, Pro mode | Claude Opus 5.5 (Anthropic) | 3 Oct 16:37 | null |
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Open Mathematics proof checks
23 proof tasks at graduate and research level, each with the proof an AI model drafted and the verdict and notes of a second AI model that checked it step by step. The tasks come from filling gaps in the Open Mathematics Courses (https://kokunoyumeto.github.io/open-math-courses-public/): operator algebras (injective factors, modular theory, KMS states, derivations), schemes and monoid schemes over the field with one element, Chevalley group schemes over the integers, and étale cohomology.
task: the task as given, with the statement to prove and the lessons and sources it could use.proof_draft: the draft, written by GPT-6 Astra (OpenAI), ChatGPT web, Pro mode, October 2026.verdictandcheck_notes: the check by Claude Opus 5.5 (Anthropic): which steps were verified against which results, errors and gaps found (in the draft and in the sources), and the conclusion.- Also:
id,title,area,received(date),continues(for answers that continue an earlier one).
Verdicts: correct: 14, complete: 6, conditional or partial: 3. Several drafts are complete only relative to cited results, and the notes say which. One task's target statement is shown to be false (P15).
These are checks by AI models, not by human referees. The lessons a task refers to are part of the courses' website; they are not included here.
Load
from datasets import load_dataset
checks = load_dataset("KokunoYumeto/open-math-proof-checks", split="train")
row = checks[0]
print(row["title"], "|", row["verdict"])
| Field | Content |
|---|---|
id, title, area |
task identifier, title and field |
task |
the task as given: the statement to prove and the lessons and sources allowed |
proof_draft |
the model's answer: Markdown with TeX formulas |
verdict |
the checker's conclusion in one line |
check_notes |
what was checked step by step, against which results, and the errors and gaps found |
drafted_by, checked_by, received |
the two models and the date |
continues |
for answers that continue an earlier answer to the same task |
Related
- The lessons, as a CC0 dataset: https://huggingface.co/datasets/KokunoYumeto/open-math-courses
- The website: https://kokunoyumeto.github.io/open-math-courses-public/
Licence
All text is marked CC0 1.0 (public domain).
Citation
@misc{open_math_proof_checks,
title = {Open Mathematics proof checks},
year = {2026},
howpublished = {\url{https://huggingface.co/datasets/KokunoYumeto/open-math-proof-checks}}
}
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