TOME-Math Turns University Textbooks into Structured Proof Supervision

Introduction

Much of the mathematical data used to train language models is presented as a collection of independent questions and answers. That format is convenient for large-scale training, but it removes information that matters in mathematics: which definitions support a proposition, how a proof is decomposed into intermediate obligations, and how a textbook gradually moves from basic concepts to more advanced arguments. TOME-Math treats university mathematics textbooks as structured sources of proof supervision rather than as text to be flattened into isolated examples.

What the corpus contains

• Scale and coverage: The corpus is built from 2,118 university-level mathematics textbooks and contains approximately 380,000 structured proof-supervision records.
• Curricular order: Records preserve chapter progression instead of treating every example as an unrelated training item.
• Evidence structure: Proposition–proof links, dependency references, intermediate proof obligations, and source provenance are retained.
• Curriculum experiments: The authors use these signals to study curriculum learning across several mathematical subjects.
• Scaling analysis: The work examines how performance changes with the amount of supervision and the breadth of source material.
• Reported outcome: At full scale, fine-tuning produces a 16.7-percentage-point gain over the base model on the unweighted mean across held-out evaluations and established public benchmarks. In aggregate, it also exceeds the three public instruction-data baselines included in the comparison.

Why organization may matter

A proof is not simply an answer attached to a question. In a textbook, definitions lead to lemmas, lemmas support propositions, and earlier results become tools for later arguments. Randomly mixing all proof records can preserve some local language patterns while obscuring the dependency graph that makes the mathematics intelligible. TOME-Math makes parts of that graph available during training, bringing the learning setup closer to the way mathematical knowledge is presented and acquired.

The project also provides a way to separate data quantity from data organization. If two training collections contain similar kinds of supervision but differ in ordering and dependency information, their results can reveal whether structure itself contributes to reasoning. Still, the available material does not establish that the reported 16.7-point improvement comes entirely from curriculum ordering. More supervision, broader sources, and structured records may all contribute to the result.

Implications and open questions

The central implication is that scaling mathematical supervision does not have to mean generating ever more disconnected exercises. Textbooks already encode concept networks, proof steps, and a progression of difficulty. Converting that material into machine-readable records could shift training research from answer coverage toward evidence structure and proof dependencies.

There are also important qualifications. The summary does not provide detailed results for each subject, benchmark, or curriculum strategy, and the corpus is announced as forthcoming rather than already released. Reproducibility will therefore depend on the eventual dataset, its licensing terms, quality controls, and documentation. Future work will need to test whether textbook-structured supervision transfers to unfamiliar mathematical tasks and whether its benefits persist across models and training procedures.

Source: Hugging Face Daily Papers