KokunoYumeto/open-math-proof-checks

Open Mathematics: arguments, reviews and corrections

This dataset preserves mathematical reasoning as it was displayed in AI-assisted work: the initial problem, the attempted argument, the reasons for questioning it, the development of a correction, and the checks that follow. It also preserves objections that turn out to be mistaken and the reasoning that changes a reviewer’s mind. The aim is to teach how an argument is developed and assessed, alongside the eventual answer.

The companion Open Mathematics Courses contains resulting lessons. The two collections connect the development of an argument to the mathematics that was incorporated into a course. Each can be downloaded independently; exact course references support using them together.

Start with the reading guide. The viewer opens on episodes: one row per reasoning history, with its learning objective and model attributions first. Choose reasoning_stages to read the complete selected text one stage at a time, or exercise_tasks for explicitly scoped training examples. P01–P21 are identifiers for mathematical assignments, not release numbers.

What is in the collection?

There are 30 reasoning histories, containing 9,525 preserved stages, and 281 scoped teaching tasks. Subjects include operator algebras, algebraic geometry, group schemes and the S6 construction and its audits. A history may include several versions of a proof, full reviews, source comparisons, worked examples and the later course adaptation.

For example:

• Complex interpolation: the writer identifies a missing construction, develops the proof and revises its display. The full reviews then distinguish a missing boundary-continuity hypothesis from an unattained supremum, and check the mathematical repair. Follow the writing, review and worked correction.
• Finite-energy limits: follow the complete domain repair, its reviews and the later proof for nondense forms. Worked explanations include counterexamples and corrections to the checking process. Follow the full lesson.
• Real coercivity: follow the missing-symmetry diagnosis, the complete replacement proof and its stronger algebraic example. Failed checks and an unfinished review retain their precise scope. Follow the argument and its checks.
• Bounded multiplication and modular equations: follow the gap between a vector solution and a bounded multiplier, the graph-cutoff argument, explicit norm completeness and the real-subspace continuation. The full proof editions retain exact examples, failed checks and the normalization needed by the integral step. Work through the arguments and checks.
• The full fundamental theorem: follow a valid restricted argument into the stronger integrable-derivative theorem, with the pointwise majorant, complete proof and both boundary examples. Preserve the original dependency-check failure and its guarded repair. Work through the reasoning.
• Original modular proof: follow four local repairs through full manuscript versions, independent reviews and a withdrawn objection. The later course connections distinguish retained text from rewritten arguments. Read the proof and review history.
• Gaussian reasoning and checks: follow the incoming web argument, Claude’s limited review, Sol’s complete supplement, Astra’s analytic rewrite and the independent review of that rewrite. The sequence preserves sharpness examples, mathematical diagnoses, failed checks and their repairs, with links to both course lessons. It also follows a reported audit defect into the manager’s investigation and corrected successor, distinguishing provenance checks from proof review. Read the full reasoning history.
• Quadratic-form polarization: follow the full converse proof, the writer’s explanation, its independent review and the repair of an older evidence record. Worked explanations cover the complex sign, infinite energies and why unchanged proof text can still depend on changed assumptions. Follow the argument and checks.
• Uniqueness of the injective factor (P04): a counterexample attacks a stronger claim than the proof requires. The exchange explains the quantifier distinction, corrects the review, and revisits a passage that had been overlooked.
• Toric gluing (P02): a general proposition needs an evaluation-compatibility hypothesis. The answer develops a counterexample without it and proves that the intended toric charts satisfy the corrected condition.
• Invariant dimensions (P08): a null set can meet every orbit. The review develops that obstruction, the writer constructs invariant dimension strata, and the reviewer checks the repaired argument.
• S6: the record connects cited research, the surviving original construction, independent audits, revised assessments and course proofs. It includes reviewers withdrawing or narrowing objections after examining the actual argument. Follow this larger chain, or read the analytic development from its initial missing hypotheses through evolving arguments, checks and ten resulting course lessons.

All 30 histories remain incomplete as records of their original development. Some initial drafting conversations and intermediate events have not been recovered. A complete delivered proof is preserved in full, but it is not represented as the missing conversation that produced it. Individual tasks can be useful within a narrower, stated scope even when the surrounding history is incomplete.

Models and evidence

Each stage distinguishes the author of the mathematical text from a later reader, reviewer, adapter or curator. Recorded contributors include GPT-6 Astra — Pro mode (ChatGPT web), Claude Opus 5.5 — Max, and Codex sessions with GPT-5.6 Sol, GPT-6 Sol, GPT-6.1 Sol or GPT-6 Astra — Ultra, where the source supports those assignments. Earlier S6 materials have qualified project-level attributions; exact model identifiers remain unresolved for some passages.

Pro mode is recorded as pro in the ChatGPT web namespace. It is not labelled “effort unknown” merely because the native generating conversation is missing, and it is not equated with Codex Ultra or a numerical API effort. A received paste establishes delivery; contemporary credits can establish its attributed model and mode. Neither supplies an absent native generation event. Read the attribution policy.

Choose a data view

View	Contents and intended use
episodes — opens by default	30 full reasoning histories, including source text, annotations, model evidence, explicit gaps and course connections.
reasoning_stages	The same histories as 9,525 ordered stages, with readable explanations and complete selected text. An inspection view, not an instruction to imitate every stage.
exercise_tasks	281 scoped tasks for proof, critique, correction, counterexamples and reassessment. Each specifies its context, target, evidence and verification limits.
provenance	141 records of course authorship declarations and separately evidenced contributions.
proof_drafts	The earlier 23 problem-and-proof records, previously named default. Select proof_drafts to load those same records.
artifacts	The earlier 27 document and review records. trajectories remains a compatibility alias; it does not imply recovered development.
legacy	An earlier export retained as evidence, with known extraction errors. Do not use it as an unqualified source of correct mathematics.

The Hub views use Parquet with a consistent schema. The complete original download structures remain available as episodes JSONL, tasks JSONL and provenance JSONL. The JSONL loader handles long records without truncation. The selection index reports task lengths, repeated targets and shared source spans.

Training use and verification limits

Historical mistakes belong in explicitly labelled context. The supplied task views assign context weight 0 and selected target weight 1; a trainer must implement those masks. A critique may quote a false formula to explain its failure. Do not treat such quotations, unsuccessful attempts or all archived tool output as positive proof demonstrations.

Preserve complete arguments and their needed prerequisites when selecting tasks for a model’s context budget. Related histories and their course lessons share a split family; keep that family together across both datasets and choose sampling weights deliberately so repeated proofs do not dominate. This public collection is not a clean held-out evaluation benchmark.

The training design explains the primary research informing these choices. The collection has not been evaluated in a training experiment, and it does not establish that one mixture or presentation is optimal. Mathematical reviews, exact source comparisons and bounded computational checks each have their stated scope; they are not blanket formal or human certification.

Connection to the course dataset

The course links identify 267 relationships using pinned dataset revisions, lesson identifiers, passage locators and content hashes. They distinguish a copied passage, an adapted proof, a cited dependency and documented source ancestry. A correction to one passage does not establish authorship or correctness of the whole lesson. Missing or damaged course exports remain explicitly marked.

The reverse course index maps every course relationship back to its exact reasoning episode and preserves the complete relationship qualifications. The joining utility verifies downloaded lesson text and carries relationship-specific qualifications into joint training. The contribution records retain the course’s authorship statement alongside the evidence about who wrote, checked and adapted the relevant mathematics.

Source preservation and licence

Original displayed explanations remain separate from later teaching annotations. Public source links, event locators and hashes make those boundaries inspectable. Missing reasoning is labelled; hidden or encrypted internal model state is not included. Mathematical arguments, reviews and corrections retain their individual source evidence and stated scope.

This dataset is dedicated to the public domain under CC0 1.0.