A textbook-based corpus links math proofs and chapters
Scaling Mathematical Proof Supervision from Structured Textbooks · arXiv:2610.40394
In plain words
Math textbooks teach ideas in an order: definitions and theorems build on earlier results, and proofs sometimes leave steps unstated. TOME-Math is a collection of textbook-based proof questions that keeps those links and adds questions about missing steps.
In tests with one base language model, training on TOME-Math gave a higher average score across six math evaluations than the untrained model and three other training collections. Separate tests also found that using textbook links to retrieve proof information helped on held-out textbook problems. These results concern the tested model and tasks.
The research
Math learning data often separates a problem from the ideas and proof steps around it. Preserving those connections could change what a training collection captures and what a model can retrieve while solving a problem.
TOME-Math tests that possibility through both model training and a separate, fixed-budget retrieval study.
TOME-Math preserves textbook chapter and proof links in a math supervision corpus, then tests how those links affect model training and evidence retrieval.
For the main comparison, researchers trained the same base model for one epoch on TOME-Math or one of three public math collections, then scored it alongside the unmodified model on six evaluations.
TOME-Math’s unweighted mean pass@1, an estimate of the correctness rate for a model’s first sampled answer, was 45.83%, versus 29.17% for the base model and 41.10% for the strongest public-corpus mean.
The evaluations score either proof correctness or final-answer correctness. TOME-Math ranked first on five and second on AIME 2026, so its higher combined mean reflects performance across this defined mix of tasks.
The corpus comparison tests one model and a defined set of textbook problems and public tests. Because the training corpora differ in several respects, that comparison alone cannot show that textbook structure caused the aggregate advantage.
What the evidence shows
- Keep textbook knowledge linked
TOME-Math is a proof-supervision corpus built from textbooks, designed to retain chapter order and links between propositions and proofs. It also records dependencies on earlier propositions and turns missing proof steps into explicit tasks.
That structure addresses a gap in supervision made from isolated problem–solution pairs, which does not preserve how textbook material is organized.
- Compare training outcomes
The shared training setup makes the corpus comparison informative about how these training collections perform under the same model initialization and evaluation.
TOME-Math has the highest unweighted average across the six evaluations, though the comparison cannot isolate textbook structure as the cause because the collections differ in several respects.
- Read the average in context
The six-evaluation mean combines different tasks: some score proof correctness and others score final-answer correctness. TOME-Math ranked first on five evaluations and second on AIME 2026.
The result therefore shows an advantage across this particular mix of tests, while the separate scores and their scoring methods matter for interpreting what that average represents.
- Test structure at answer time
The retrieval comparison tests whether selecting evidence through textbook structure helps a fixed model solve held-out textbook problems.
The complete retrieval condition performed best across the six subjects, supporting the value of combining linked propositions with explicit proof gaps in this tested setting.
- Probe order and coverage
The paired schedule comparisons test chapter direction while keeping Gap QA first, and test Gap-first ordering while keeping chapter progression forward. Forward–Gap performed strongest across the subject and budget combinations, supporting contributions from both ordering choices in these training setups.
A separate matched-token comparison found that using a broader source pool improved scores across subjects, suggesting that source coverage can matter independently of token volume.
Deep
- Original paper authors
- Youqing Fang, Zhouqi Hua, Yinhao Tang, Yuzhe Gu, Lingkai Kong
- Submitted to arXiv
- License
- CC BY 4.0