Scaling Mathematical Proof Supervision from Structured Textbooks

Project page: https://tome-math.github.io/

Introduction

Mathematical reasoning in large language models has advanced rapidly, driven in part by improvements in training data and in part by advances in reasoning-time computation [1, 2, 4, 10, 13, 15, 21, 27, 28, 29, 37, 38, 42]. On the data side, recent efforts have expanded mathematical supervision to millions of QA pairs (i.e., Question-Answer pairs) and in doing so have revealed that properties such as question diversity, solution format, and teacher-model quality significantly shape downstream performance [18, 31, 39, 40]. These findings suggest a broader principle: The value of mathematical supervision is not determined by the number of training examples alone, but also by how those examples are constructed and organized.

Despite this, the dominant paradigm for mathematical supervision remains the isolated problem–solution pair. Whether the original knowledge comes from competition archives, grade-school word problems, web exercises, or synthetic augmentation, the resulting training data typically takes the uniform shape of problem $\to$ solution [7, 14, 39]. This representation is convenient to scale, but it does not explicitly leverage the organizational structure that many source materials already possess.

Mathematical textbooks are a notable case in point. A textbook is not merely a collection of solved problems, but a carefully structured body of knowledge. Definitions introduce core concepts, lemmas and theorems develop them, proofs establish their validity, and exercises reinforce understanding. These elements are organized within a chapter hierarchy that reflects prerequisite relations, conceptual dependencies, and pedagogical progression. Prior work has explored textbooks as a source of mathematical pretraining data, problem–solution pairs, and proof supervision [3, 16, 19, 30, 33, 35, 41]. However, textbooks are often treated primarily as sources of mathematical content, leaving their native structure underexplored. In particular, it remains unclear how to scale natural-language proof supervision while preserving chapter order, proposition dependencies, source locations, and exercises in a unified framework.

To address this gap, we introduce TOME-Math (Textbook-Oriented Mathematical Evidence), a large-scale proof-supervision resource constructed from 2,118 mathematics textbooks. Central to TOME-Math is the notion of structured mathematical supervision, which preserves structure at three levels. First, hierarchical structure retains the native organization of textbooks, including chapters, propositions, and their ordering. Second, dependency structure connects theorem-level proof questions with the intermediate proof obligations derived from gaps in their proofs. Third, curriculum structure organizes supervision according to textbook order and proof granularity, enabling controlled training over related examples rather than independent QA pairs. Following these principles, TOME-Math contains 380,292 proof-oriented QA pairs spanning 29 mathematical subjects. Figure 1 provides an overview of the construction and organization of the resulting corpus.

We further validate the effectiveness of TOME-Math through supervised fine-tuning (SFT) on a 30B base model [38]. Its retained structure enables controlled studies of how the organization of supervision affects mathematical reasoning. Rather than treating scale as the only variable, TOME-Math allows us to examine the roles of retrieval structure, curriculum order, and source coverage under controlled settings. Empirically, TOME-Math SFT improves the six-benchmark average from 29.17% to 45.83%, while further experiments show consistent benefits from structure-guided retrieval, textbook-aligned curricula, and broader source coverage. Our contributions are:

• We introduce TOME-Math, a large-scale proof-supervision resource constructed from 2,118 mathematics textbooks. Through a structure-preserving pipeline, it produces 380,292 Concept and Gap QA records while retaining chapter progression, proposition–proof links, dependency references, source provenance, and a separate inventory of textbook-authored exercises.

• We demonstrate the overall effectiveness of TOME-Math through full-corpus fine-tuning. On book-held-out textbook exercises and established public benchmarks, TOME-Math achieves the best performance on five of six evaluations, improving the unweighted six-benchmark mean from 29.17% to 45.83% over the base model and outperforming all three public instruction-data baselines in aggregate.

• We systematically investigate how textbook structure contributes to mathematical reasoning at inference and training time. Structure-guided retrieval improves proof-generation pass@1 from 13.25% to 17.38% under a matched context budget; across six subjects and three data scales, forward textbook curricula consistently outperform shuffled and reverse schedules; and broader source coverage yields further gains under a fixed training-token budget.

Resource	Size	Source	Supervision	Metadata / structure
Mathematical instruction data
MATH [2021]	12.5K QA	Competitions	Solutions	Subject, difficulty
MetaMathQA [2023]	395K QA	GSM8K, MATH	Solutions	Seeds, augmentation types
OpenMathInstruct-2 [2024]	14M QA	GSM8K, MATH	Solutions	Source labels
NuminaMath-CoT [2024]	∼860K QA	Exams, web, synthetic	Solutions	Source labels
Textbook and knowledge-informed resources
MathPile [2024]	9.5B tokens	Books, papers, web	Text	Sources, file paths
CollegeMath [2024]	4,099 QA	9 textbooks	Short answers	Book/section/exercise IDs
Proof resources and evaluation
NaturalProofs [2021]	32K proofs	2 textbooks + web	NL proofs	Reference graph
Proof Net [2023]	371 examples	Textbooks, Putnam	NL + Lean stmt.	Book/exercise IDs
Lean-GAP [2026]	430 problems	1 textbook	Lean statements	Section/exercise IDs
AlgGeoTest [2025]	456 questions	Stacks Project	Proof judgments	Statement tags
TOME-Math (ours)	380,292 QA	2,118 textbooks	NL + local proofs	Chapter order, refs, provenance

Related Work

Mathematical instruction data. GSM8K and MATH provide human-authored problems and solutions for grade-school and competition mathematics [7, 14]. MetaMathQA and OpenMathInstruct-2 scale these seeds through synthetic augmentation [31, 39], while NuminaMath-CoT aggregates diverse sources with chain-of-thought solutions [20]. Their primary supervision unit is a problem–solution pair, accompanied by source, seed-question, or difficulty metadata. Table 1 contrasts this organization with the textbook hierarchy and proposition links retained by TOME-Math.

Textbook and knowledge-informed resources. OpenWebMath and MathPile provide mathematical pretraining text, with MathPile incorporating textbooks alongside papers and web sources [25, 33]. CollegeMath extracts textbook exercises and short answers, while MathScale synthesizes QA using a graph of topics and knowledge points [30]. TheoremQA evaluates theorem application across STEM subjects [6]. These efforts motivate retaining both textbook organization and links to mathematical statements when constructing proof supervision.

Proof resources and evaluation. NaturalProofs preserves statements, proofs, and a reference graph [35, 36], while NaturalProofs-Gen and NaturalProver study reference-grounded proof generation. ProofNet and Lean-GAP connect textbook mathematics to formalization [3, 19], while FormalRL’s uproof provides 5,273 proof problems from 14 textbooks for autoformalization evaluation [16]. Proof2Hybrid’s AlgGeoTest studies proof-derived judgments, and the Open Proof Corpus studies automated assessment of human-evaluated model proofs [8, 26]. Building on these directions, TOME-Math combines coverage of 2,118 textbooks with self-contained local proof obligations, chapter order, and proposition references, supporting controlled studies of source coverage, proof-target format, and training order.

TOME-Math

Data Construction

Pipeline overview. We construct TOME-Math through a structure-preserving pipeline that converts raw textbook PDFs into two complementary forms of proof supervision: Proof QA derived from explicit proposition–proof units and Gap QA derived from missing proof obligations in Proof QAs. The source collection is selected against university course syllabi and reviewed by domain experts, yielding 2,118 input PDFs after canonicalization and deduplication (as shown in § 3.2). The pipeline generated the full TOME-Math dataset of 380,292 QA pairs, comprising 174,431 Gap QA pairs and 205,861 Proof QA pairs.

Document parsing and structured chapter extraction. DeepSeek-OCR [34] converts each PDF into multimodal Markdown together with page images. A page-wise gpt-oss-120b [23] pass separates the table of contents and main body from covers, prefaces, and other front matter, then reconstructs a hierarchical table of contents. Conditioned on this hierarchy, a state machine reads the body in order, judges whether each page belongs to the current chapter, the next chapter, or contains an internal boundary, and retries ambiguous assignments before updating the chapter state. This produces an ordered set of chapter-level documents while preserving the textbook hierarchy. A second page-wise pass separates expository core concepts from textbook exercises within each chapter. The exercises are retained as an independent reservoir, but are excluded from the Proof QA generation path.

Proof QA Extraction and Gap QA construction. For each core concept chapter, we use gemini-3.1-flash-lite-preview [12] to extract theorem–proof, lemma–proof, and definition–proof units and retrieve the earlier propositions needed to interpret each proof. These source-grounded units form the Proof QA stream. We then use gemini-3-flash-preview with thinking enabled [11] to inspect each proof for gaps (i.e., implicit or missing reasoning steps). When a substantive gap is detected, gemini-3.1-flash-lite-preview [12] rewrites the missing obligation as a self-contained proposition and completes its proof, producing the Gap QA stream. This structured construction preserves explicit textbook proofs while adding targeted supervision for intermediate reasoning that the source leaves unstated.

Validation and quality control. Quality control is applied at both the structural and semantic levels. The structural checks validate the table-of-contents hierarchy, ambiguous page boundaries, question continuity across pages, question–answer alignment, and standalone completeness. For proposition extraction, a source-conditioned gpt-oss-120b judge audits candidates along five axes: completeness, accuracy, \LaTeX formatting, self-containment, and atomicity. Accuracy is treated as critical: invented proof steps, altered mathematical relations, or other content not supported by the source are failure cases rather than stylistic defects. Full module logic, intermediate artifacts, model-allocation details, and the judging rubric are given in Appendix § 7.

Characterization

TOME-Math comprises 380,292 QA records from 2,118 canonical textbooks across 29 subject categories, summarized in Figure 3. Appendix § 8 provides the detailed corpus characterization, including subject and language coverage, task composition, structural complexity, confidence dynamics, quality, and provenance.

Generation Composition. The corpus combines a Proof QA stream of 205,861 records and a Gap QA stream of 174,431 records, drawing on disjoint model assignments (Appendix § 8). gemini-3-flash-preview-thinking [11] produces 88.9% of Proof QA records, gpt-5-mini [22] produces 53.7% of Gap QA records, and the Gap QA stream retains 174,431 of 918,235 generated candidates, a yield of 19.00%.

Model-quality trade-off. We evaluate model-quality trade-off along six dimensions, and combine gemini-3-flash-preview-thinking (Flash Thinking) for primary extraction and auditing with gemini-3.1-flash-lite-preview (Flash Lite) for reference-semantic parsing, balancing quality and cost. See more details in Appendix § 7.§ 6.

Provenance and Evaluation Units. Each record carries book, chapter, and source location metadata, with proposition identifiers and resolved references for Proof QA records. We define two evaluation scopes at the book level, held out exercises for item level generalization and held out books for source level generalization, both respecting duplicate groups identified by global content hash.

Evaluating TOME-Math

Experimental Setup

Base Model. We use Qwen3-30B-A3B-Base [38] as the backbone for all experiments. All SFT variants start from the same checkpoint, and the unmodified checkpoint serves as the base-model baseline.

Evaluation. To obtain a sensitive view of how structured textbook supervision affects learning, we focus on the six core subjects with the largest usable textbook collections in our corpus: abstract algebra, differential geometry, discrete mathematics, mathematical analysis, probability, and topology. We construct TOME-LONGFORM from their textbook-authored exercises: 300 proof-heavy problems from held-out books, balanced across subjects and the earlier and later halves of each book. To enforce book-level separation, we reserve 64 source books for evaluation and exclude all 15,661 QA records associated with those books from the complete 380,292-record TOME-Math corpus. This leaves 364,631 examples in the actual training split. The evaluation problems are selected separately from the native-exercise reservoir of the held-out books. We sample 16 solutions per problem and score them with a reference-guided step verifier; each solution is reviewed eight times by gpt-5.6-luna [24] and is accepted only if no review identifies a substantive error. For complementary coverage beyond long-form proofs, TOME-OBJECTIVE contains 145 multidisciplinary objective problems scored by final-answer equivalence. We additionally evaluate on MiniF2F [43], ProofNet [3], MATH-500 [14], and AIME2026 [9]. Full construction and scoring details are given in Appendix § 11.

Full-Corpus Comparison

We first present the overall effectiveness of the final TOME-Math dataset, before isolating its structural components in § 5. Table 2 compares the resulting 364,631-example TOME-Math training split with three public mathematics SFT corpora at their available scales. All fine-tuned models start from Qwen3-30B-A3B-Base and use the same fine-tuning setup, differing only in their supervision corpus; each model is trained for one full epoch over its respective corpus, and the unmodified checkpoint is included as a reference. Complete training configurations are provided in Appendix § 13.§ 2.

SFT supervision corpus	TOME-LF	TOME-Obj.	MiniF2F	ProofNet	MATH-500	AIME-26	Avg.
None (Qwen3-30B-A3B-Base)	13.25	32.93	34.45	17.14	68.80	8.44	29.17
OpenMath1 (1.82M)	20.16	33.86	22.87	19.25	70.40	8.21	29.13
MetaMath (395k)	28.73	34.66	40.70	22.55	71.20	9.98	34.64
MathInstruct (262k)	<u>35.87</u>	<u>40.69</u>	<u>52.05</u>	<u>32.93</u>	<u>73.05</u>	11.98	<u>41.10</u>
TOME-Math (ours; 364.6k)	43.86	43.96	57.43	42.98	76.55	<u>10.19</u>	45.83

Aggregate performance. TOME-Math achieves the best result on five of the six benchmarks and ranks second on AIME 2026. Its largest margins over the strongest public baseline appear on the proof-oriented TOME-LONGFORM and ProofNet evaluations, while its advantages on TOME-OBJECTIVE, MiniF2F, and MATH-500 show that the benefit extends beyond its own evaluation distribution and is not confined to long-form proof generation. Averaged uniformly across the six reported scores, TOME-Math improves the base-model mean from 29.17% to 45.83%, a gain of 16.66 percentage points (16.7 when rounded to one decimal).

Effect of supervision corpus. The ranking is not explained by corpus size alone. OpenMath1 contains 1.82M examples, approximately $4.8 \times$ the size of TOME-Math—yet TOME-Math outperforms it on all six benchmarks. It also exceeds the similarly sized MetaMath corpus throughout and improves consistently over the pretrained backbone. These results demonstrate the overall utility of TOME-Math and suggest that the source and form of supervision matter in addition to raw example count. Because the corpora differ in several respects, the comparison does not by itself isolate textbook structure as the causal factor; we examine that structure through controlled retrieval and training studies in § 5.

Leveraging Textbook Structure

Beyond providing a large source of proof supervision, TOME-Math exposes structural signals—chapter progression, proposition dependencies, and source provenance—that can be used explicitly at inference time and during training. This section investigates when these signals improve mathematical reasoning. We first evaluate structure-guided retrieval without updating model parameters (§ 5.1). We then study whether textbook order provides an effective curriculum under controlled budgets (§ 5.2), and examine source breadth together with transfer to held-out books from other mathematical fields (§ 5.3). Finally, we examine how these structural choices contribute to the full-corpus TOME-Math training recipe (§ 5.4).

Training-Free Structure-Guided Retrieval

Question and design. Can textbook structure identify useful evidence for mathematical problem solving? We test five paired context policies on all 300 TOME-LONGFORM problems using a frozen Qwen3-30B-A3B-Base model: (i) Cʟᴏsᴇᴅ uses no external context; (ii) Rᴀɴᴅᴏᴍ Sᴀᴍᴇ-Bᴏᴏᴋ samples uniformly from the exercise’s source book; (iii) Lᴏᴄᴀʟ Cʜᴀᴘᴛᴇʀ draws only from the exercise chapter and its immediate predecessors; (iv) TOME (Pʀᴏᴘs Oɴʟʏ) retrieves proposition–proof records using chapter position and recovered dependencies; and (v) TOME (Pʀᴏᴘs + Gᴀᴘ) extends the same policy to include Gap QA.

All context-augmented conditions receive the same context-token budget. They also share the same backbone, prompt, decoding configuration, and reference-guided verifier; the target exercise and material overlapping its reference answer are removed from every retrieval pool. The resulting comparison asks how effectively each policy selects a fixed amount of same-source evidence, rather than whether additional test-time tokens alone help. Appendix § 14 gives the complete retrieval, leakage-control, generation, and scoring protocol.

Results. The results show a clear progression as increasingly structured evidence is introduced. Starting from 13.25% pass@1 in the CLOSED setting, RANDOM SAME-BOOK and LOCAL CHAPTER reach 14.00% and 14.71%, respectively. Retrieving proposition–proof records with textbook structure further improves performance to 15.54%, while augmenting this inventory with Gap QA yields the best overall score of 17.38%, a 4.13-point gain over closed-book generation. In particular, Gap QA contributes an additional 1.84 points over proposition-only retrieval, showing that explicit local proof obligations provide complementary evidence beyond textbook propositions. Among the four context-augmented conditions, the shared token budget shows that their performance differences arise from evidence selection rather than context quantity.

The advantage of the complete TOME (PROPS + GAP) policy is consistent across all six subjects, with improvements of 1.75–6.00 points over CLOSED. Mathematical analysis and topology show the largest gains (+6.00 points each), while every other subject also improves. By contrast, random same-book or local-chapter context occasionally leaves performance unchanged or lowers it, indicating that access to nearby textbook material alone is not sufficient. Together, these results establish the retained textbook structure as an effective inference-time evidence index and show that proposition context and explicit proof obligations work best in combination.

Curriculum Learning Across Subjects and Data Scales

The retrieval study shows that textbook structure can guide the selection of useful evidence at inference time. We next ask whether the same organization can also improve learning when it is used to order supervision, and whether any benefit persists across subjects and data scales.

Setup and comparisons. We evaluate textbook-aligned curricula across six subjects and three per-subject target-token budgets: 0.75M, 1.8M, and 2.4M. Within each subject–scale cell, four schedules reorder the same frozen QA set: SHUFFLE, FORWARD–GAP, REVERSE–GAP, and FORWARD–MIX. Here, Gap QA makes a locally omitted proof obligation explicit. FORWARD–GAP versus REVERSE–GAP isolates chapter direction under a fixed Gap-first policy, while FORWARD–GAP versus FORWARD–MIX isolates Gap-first ordering under a fixed forward progression; comparison with SHUFFLE measures their combined effect. All schedules share the same examples, token budget, initialization, and optimization settings, and are evaluated on the same held-out book exercises. Appendix § 15 details the normalized chapter stages, schedule construction, nested data scales, and order-preserving training controls.

Overall effect of scale and curriculum. Figure 4 shows that every fine-tuned condition outperforms the shared base checkpoint in all six subjects, even at 0.75M tokens, and that every schedule improves with scale in each subject. Across the 0.75M, 1.8M, and 2.4M budgets, macro-average pass@16 rises from 41.04% to 42.52% and 43.21% for SHUFFLE, and from 45.38% to 48.98% and 50.93% for FORWARD–GAP. Thus, both supervision scale and structured ordering improve solution coverage.

Effect of chapter direction and Gap-first ordering. FORWARD–GAP is strongest in all 18 subject–scale cells, and its macro-average margin over SHUFFLE widens from 4.34 points at 0.75M tokens to 6.46 at 1.8M and 7.72 at 2.4M. Controlled comparisons attribute most of this benefit to chapter direction: relative to REVERSE–GAP, the forward schedule gains 5.65, 7.41, and 8.50 points across the three scales. Holding forward progression fixed, Gap-first ordering adds a smaller but consistent 2.37, 1.91, and 1.53 points over FORWARD–MIX. Thus, forward progression is the primary contributor, with Gap-first ordering providing a complementary gain. The same ordering across all 18 subject–scale cells shows that these benefits persist across mathematical domains and supervision budgets.

Source Breadth and Cross-Subject Transfer

Section 5.2 shows that performance improves as the subject-specific training sets expand, but token volume and source count grow together. We therefore ask whether broader textbook coverage helps at a fixed token budget and whether subject-specific supervision generalizes across fields. Figure 5 summarizes these breadth and transfer studies.

Source breadth. Figure 5(a) compares the 8-book NARROW–DENSE condition with BROAD–SPARSE, which distributes the same 0.75M-token FORWARD–GAP budget across 16 books while holding all other settings fixed. BROAD–SPARSE improves pass@16 in every subject by 1.6–4.8 points (3.3 on average), with the largest gains in topology (+4.8), differential geometry (+4.4), and discrete mathematics (+3.6). Thus, source breadth is a distinct scaling axis: the same supervision budget is more effective when spread across twice as many books. Together with the monotonic gains in § 5.2, this result shows that textbook supervision benefits from both greater volume and broader coverage. Table 12 details the source pools and QA counts.

Subject	0.75M tokens Books	0.75M tokens QA	1.8M tokens Books	1.8M tokens QA	2.4M tokens Books	2.4M tokens QA
Discrete mathematics	8	2,273	16	5,258	24	6,847
Mathematical analysis	8	2,387	16	5,809	24	7,630
Abstract algebra	8	2,235	16	4,899	24	6,831
Probability	8	2,062	16	4,694	24	6,230
Differential geometry	8	1,899	16	4,033	24	5,537
Topology	8	1,930	16	4,877	24	6,571
Total	48	12,786	96	29,570	144	39,646

Cross-subject generalization. Figure 5(b) evaluates each subject-specific 0.75M-token checkpoint on 50 exercises from every other field, normalized by the target specialist’s in-domain score, $T_{s \to t} = P_{s \to t}/P_{t \to t}$. The macro-average transfer coefficient is 0.92. Topology (0.98) and differential geometry (0.97) have the strongest source-side means, and eight of the 30 ordered pairs exceed 1.0; both fields reach 1.29 on probability targets. This strong off-diagonal performance shows that subject-specific supervision develops reusable proof and reasoning patterns rather than benefits confined to its training field. It also supports multidisciplinary training at full scale. Full results are in Appendix § 16.

Taken together, the matched-budget breadth gains and cross-subject transfer connect the controlled studies to the full TOME-Math corpus. Scaling to more books adds source coverage without increasing the token budget, while scaling across subjects supplies supervision that generalizes beyond its domain. These complementary effects explain the effectiveness of the broad, multidisciplinary corpus beyond example count alone.

Full-Corpus Ablation of the TOME-Math Training Recipe

Finally, we ablate curriculum order, Gap-first scheduling, and Gap QA from the full FORWARD–GAP recipe and report pass@1 (%) for all variants. All variants use the book-held-out training split described in § 4.

Component contributions. Table 4 shows that every ablation lowers pass@1 across all six benchmarks. On the two TOME benchmarks, replacing Gap-first with within-stage mixing costs 3.74 and 2.59 points, while global shuffling costs another 0.25 and 0.95 points. Thus, Gap-first is the primary ordering signal, complemented by forward chapter progression. Removing Gap QA causes the largest drops on the four proof benchmarks (5.05–6.77 points), confirming the value of explicit local proof obligations. The complete recipe also leads on MATH-500 and AIME-26, extending the gains beyond proof-focused evaluations.

Variant	TOME-LF	TOME-Obj.	MiniF2F	ProofNet	MATH-500	AIME-26
w/o Gap QA (FORWARD)	38.72	38.91	50.73	36.21	73.84	9.12
w/o curriculum (SHUFFLE)	39.87	40.42	53.76	39.12	75.22	9.58
w/o gap-first (FORWARD–MIX)	40.12	41.37	55.61	41.02	75.88	9.93
TOME-Math (FORWARD–GAP)	43.86	43.96	57.43	42.98	76.55	10.19

Structural synergy. The ablations reveal a mutually reinforcing relationship between TOME-Math’s corpus construction and training strategy. Gap QA introduces explicit proof obligations, Gap-first scheduling prioritizes them within each stage, and forward progression exploits the preserved textbook order. The consistent gains across textbook-derived proof tasks, public proof benchmarks, and competition problems establish TOME-Math’s advantage as structured supervision beyond scale alone.

Conclusion

We introduced TOME-Math, a corpus of 380K proof records from 2,118 textbooks that preserves curricular and dependency structure. It improves performance across textbook-derived and public benchmarks, while controlled studies demonstrate the benefits of structure-guided retrieval, curriculum ordering, and broader source coverage. We hope TOME-Math inspires mathematical reasoning systems that learn not only from more data, but from better-organized knowledge.

TOME-Math Construction Details

This appendix expands the construction stages summarized in § 3.1. We distinguish the artifacts produced by the book-structure modules (M1–M3), exercise and answer processing (M4–M6), and the downstream proposition-extraction track.

Intermediate representation

DeepSeek-OCR produces one multimodal-Markdown representation of the complete book together with the extracted images. The automated modules materialize the following intermediate artifacts:

Artifact	Contents
toc.json	Multilevel table of contents represented by title and heading depth.
main_text.md	Textbook body after removing front matter.
images/	Merged image directory with normalized references from the Markdown files.
content/	One Markdown file for every recovered chapter.
content_pure/	Chapter text after exercise segments are removed.
exercise/	Chapter-indexed exercise text and formatted per-chapter JSONL files.
exercise_index.jsonl	Multilevel heading path for every exercise section.
answer.md	A detached answer chapter when one is present.

M1–M2: Table-of-contents and chapter recovery

M1: front matter and table of contents. Starting from the first parsed page, the structure model decides whether a page contains table-of-contents entries and whether the textbook body has begun. For contents pages it extracts pairs of the form {title, level}. Pages that are neither contents nor body are treated as cover or front matter and are discarded. Once the body is detected, a verification pass corrects invalid heading depths and removes entries that belong to prefaces or notation lists. If the model detects body text before extracting any valid contents entries, the whole module is retried, up to five attempts. M1 emits toc.json, main_text.md, and the merged images/ directory.

M2: hierarchy-conditioned segmentation. The recovered contents are traversed as a tree. Internal entries direct the traversal to their children, while leaf entries trigger a search for the corresponding body span. For each page and the current/next chapter titles, the model returns one of three states: ALL-CURRENT (the page belongs to the current chapter), ALL-NEXT (it belongs to the next chapter), or BOUNDARY (the transition occurs within the page). Boundary pages are checked a second time; a failed check is fed back to the assignment step. A successful transition advances the state machine and writes the completed chapter to content/.

M3–M6: Exercise extraction and answer alignment

M3: exercise separation. The structure model first tests whether the table of contents contains dedicated exercise chapters. When it does, those chapters are detached directly. When it does not, each content chapter is scanned for a trailing block of exercises, which is removed from the exposition. The same module constructs the multilevel exercise-title index and detects a separate answer chapter, if present. Its outputs are content_pure/, exercise/, exercise_index.jsonl, and optionally answer.md.

M4: exercise serialization. The parser reads each exercise section page by page and emits one JSON record per problem. A record contains question, answer when an answer is printed immediately after the question, and number. The final problem fragment on a page is carried into the next page before serialization so that a page break does not split a question into two records. Problem numbers are preserved for the matching stage.

M5: answer partitioning. The pipeline determines whether numbering is absolute—one number identifies a unique problem in the whole book—or relative, with numbers restarting across chapters. For absolute numbering, the answer chapter is serialized using the same page-wise procedure as M4. For relative numbering, the multilevel heading paths saved by M3 are used to partition the answer chapter before serialization.

M6: matching and filtering. Absolute-number records are joined directly on problem number. Relative-number records are matched within their recovered chapter, using the section content when necessary. Books without printed answers follow a third, unmatched branch. All branches then pass through a completeness check that removes questions whose statements depend on an unprovided theorem or on the conclusion of another exercise. The module produces a report with the number of exercises and answer coverage for each chapter.

Four-stage proposition pipeline

Each pure-content chapter is first split at Markdown headings while preserving its preamble, and the resulting sections are processed concurrently. The four semantic stages are as follows.

1. Proposition extraction. For every section, the model emits structured proposition records with type, number, label, title, content_tex, and proof_tex fields.

2. Reference resolution. A book-level proposition library is built from the extracted records. For each proposition with a proof, the model identifies both explicit \ref citations and semantic references to earlier results, storing the matches in resolved_references.

3. Proof-gap audit. Each proof is checked for unverified lemma assumptions, omitted coordinate or operator transformations, and unsupported jumps marked by phrases such as “obviously” or “clearly.” A candidate gap stores gap_statement, missing_logic, sub_proposition, and a low/medium/high difficulty label.

4. Self-contained refinement. Each candidate sub-proposition is rewritten as an independent mathematical statement by declaring symbols, injecting required definitions, and preserving the source language.

The chapter-level structured output contains the extracted propositions, resolved references, and candidate gaps. A separate Proof QA JSONL stream distinguishes original proposition records from candidate sub-propositions with an is_potential_gap flag. This schema describes the pipeline output; dataset-level acceptance, deduplication, and split policies are subsequent steps.

Proposition extraction and model allocation

The pure content chapters feed the proposition-centered training-data pipeline. The construction log compares three individual Gemini configurations and a hybrid allocation. The later cost-optimization note assigns second-stage reference semantics to the lower-cost non-thinking model and pairs it with the thinking model for the remaining reasoning-intensive operations. Because this allocation changed during pipeline development, it is versioned with each run. Table 5 reports the per-book costs recorded in the design document.

Configuration	Cost / book
Config 1: gemini-3.1-pro-preview-thinking	$40–50
Config 2: gemini-3.1-flash-lite-preview	approx. $5
Config 3: gemini-3-flash-preview-thinking	approx. $10
Config 2 + Config 3 hybrid	approx. $7

Model-Quality Evaluation and Model Selection

Evaluation Dimensions. We evaluate model quality along six dimensions:

• Completeness measures the completeness of extracted information;

• Accuracy assesses factual correctness;

• LaTeX Formatting evaluates compliance with LaTeX formatting requirements;

• Self-Containment measures whether each extracted item is independently understandable;

• Atomicity assesses whether each item represents a single, coherent unit

• C/P Ratio measures cost-effectiveness in terms of the cost of generating a single QA pair, with a higher score indicating a lower cost per pair.

Model Configurations. We compare four configurations:

• Flash Thinking refers to gemini-3-flash-preview-thinking

• Flash Lite refers to gemini-3.1-flash-lite-preview

• Pro Thinking refers to gemini-3.1-pro-preview-thinking

• Combined denotes a pipeline combining Flash Thinking for primary extraction and auditing with Flash Lite for reference-semantic parsing.

Results and Analysis. Figure 3 shows that Pro Thinking achieves strong performance in Completeness, Accuracy, LaTeX Formatting, Self-Containment, and Atomicity, but performs less favorably in C/P Ratio, indicating a higher cost per generated QA pair. Flash Lite offers a more cost-effective alternative, although its lower scores in Completeness and Self-Containment suggest potential limitations in information coverage and the independent interpretability of extracted items. Flash Thinking provides consistently strong performance across the five quality-oriented dimensions, while maintaining a reasonable cost per QA pair. Notably, the Combined pipeline preserves competitive extraction quality while improving cost-effectiveness through the use of Flash Lite for reference-semantic parsing. Overall, these results suggest that the Combined pipeline offers a favorable balance between extraction quality and the cost of generating individual QA pairs.

Extended Characterization Analysis

This appendix provides the detailed corpus characterization summarized in § 3.2. We first describe subject and language coverage and task composition, then examine quality behavior, provenance, reasoning scope, structural complexity, and confidence.

Task and Answer-Format Composition

Figure 9(a) shows that proof generation accounts for 79.0% of records, an order of magnitude above every other task type. Figure 9(b) crosses task type with answer format. Derivation and calculation binds almost deterministically to the derivation format at 87.0%, while proof generation splits between full textbook proof (51.0%) and concise final proof (41.9%).

Subject and Language Coverage

Figure 10(a)-(b) reports QA volume and distinct source book count for the fifteen most represented subjects. The ten leading subjects account for 84.4% of records. Mathematical analysis draws 66,409 records from 307 books (216 per book), while graph theory draws 24,406 records from only 25 books (976 per book), reflecting broad coverage for analysis-heavy subjects against concentration on specialized books elsewhere. Source language composition varies by educational level. All 337 high school records have mixed-language sources. Undergraduate records comprise 93,537 Chinese, 71,702 English, and 23,500 mixed-language examples; competition records comprise 24,538 Chinese (92.4%) and 2,024 mixed-language examples; and graduate records comprise 109,648 English (66.6%), 35,967 Chinese, and 19,039 mixed-language examples.

Quality Behavior at Subject and Dimension Granularity

Figure 11(a) presents subject-specific quality-tier profiles. Competition mathematics has tier A and tier C shares of 21.6% and 71.5%, respectively, while abstract algebra and topology each have tier A shares above 39%. Figure 11(b) reports results across nine automated audit dimensions. Ambiguity and question–answer consistency pass above 90% of records. Groundedness and evidence alignment assess correspondence with the source material; both pass below 50% of records and have a fail rate of 28.6%. This dimension-level breakdown provides a detailed profile of question–answer quality and source-evidence traceability to guide targeted review.

Corpus Redundancy and Generation Provenance

Figure 12(a) reports the reasons records are excluded outright. Formula broken and incomplete proof are the two leading causes, together accounting for 36,775 excluded records, with the top four reasons covering 80% of all exclusions. Figure 12(b) corroborates the dimension level asymmetry in Figure 11(b) from an independent signal drawn from human review flags. The flag no verbatim evidence covers 48.4% of the corpus, 2.7 times the frequency of the second most common flag, needs math review.

Corpus Redundancy and Generation Provenance

Figure 13(a) reports exact duplicate group sizes. Singleton records account for 99.15% of the corpus and duplicate groups account for only 0.85%, with one outlier group of size twenty one isolated from the remaining groups, all of size four or smaller. Figure 13(b) traces the generation composition introduced in § 3.2 to individual generator models. The Proof QA and Gap QA streams draw on disjoint model assignments, with gemini-3-flash-preview-thinking responsible for 88.9% of Proof QA records and gpt-5-mini responsible for 53.7% of Gap QA records.

Figure 14 traces the subject-level concentration in Section B.2 to individual sources. The single highest yield book, Jukna’s Extremal Combinatorics, contributes 3,049 records, 0.80% of the corpus, and the top twenty books are dominated by English language graph theory and combinatorics textbooks together with Chinese competition mathematics series.

Structural Complexity and Confidence Dynamics

Reasoning Scope and Subject Composition Across Level

Figure 15(a) examines how reasoning scope, the breadth of chapters and concepts a record draws on, varies with educational level. The combined share of multi step same chapter and cross chapter reasoning reaches 32.5% at the graduate level, compared with 18.7% for undergraduate and 17.7% for competition. Competition and undergraduate report similar single concept shares despite differing in solving difficulty, indicating that reasoning scope captures curriculum breadth rather than solving skill. Figure 15(b) places this finding in the context of subject coverage from Section B.2. Mathematical analysis is the only leading subject with substantial coverage at both undergraduate and graduate levels, while most other leading subjects concentrate on a single level, explaining why level and subject interact strongly in downstream evaluation splits.

Structural Difficulty and Record Length

Each record carries structural metadata (curriculum stage, structural difficulty, concept count, prerequisite depth) and a model-reported confidence score. Figure 16(a)-(Figure 16b) shows that prerequisite count correlates with estimated reasoning steps at 0.633, stronger than concept count at 0.522; the fitted surface also rises more steeply along the prerequisite-count axis, making prerequisite depth the dominant driver of reasoning-chain length. Figure 16(c) crosses confidence with curriculum stage and difficulty. Confidence falls with difficulty only at later stages, from 0.876 to 0.807 at stage four and from 0.873 to 0.788 at stage five, while stages one through three remain flat. Figure 16(d) shows that answer length correlates with confidence at −0.231, compared with −0.060 for question length, suggesting that longer derivations drive lower confidence.

Structural Difficulty and Record Length

Figure 17(a) reports the underlying joint distribution of curriculum stage and structural difficulty. A single cell at stage three, difficulty three holds 194,059 records, 51.0% of the corpus, and stage three overall accounts for 65.9% of records, indicating that structural difficulty tracks curriculum progression closely rather than varying freely within a stage. Figure 17(b) reports the joint distribution of question and answer length by quality tier. The two lengths are positively associated but retain substantial spread within every tier, so tier assignment cannot be inferred from length alone.

Confidence Distributions by Tier and Curriculum Stage

Figure 18(a) reports the marginal confidence distribution by quality tier. Mean confidence follows the order tier A at 0.944, tier B at 0.873, and tier C at 0.821, with tier A showing the tightest spread at a standard deviation of 0.030, indicating that confidence separates tier A cleanly from the lower tiers. Figure 18(b) reports the marginal distribution by curriculum stage, which follows a non monotonic pattern of 0.892, 0.865, 0.882, 0.835, and 0.840 across stages one through five, with stage three holding both the largest sample share at 66.0% of the corpus and the tightest spread.

Synthesized QA Samples

To make the pipeline output concrete, this appendix pairs each synthesized QA with the exact source paragraph it was derived from. We draw all five items from a single textbook—Diestel’s Graph Theory, one of the graph-theory books in the corpus—so that the reader can trace how the same book supplies items at every level of the L1–L5 ladder used in § 3.2. Each block below reproduces the source excerpt (grey) and the QA record (green) verbatim from the intermediate representation and the release table; only whitespace, LaTeX escaping, and page-break markers were adjusted for typesetting.

L1 – Direct fact recall (Section 12.3, Tree-decompositions)

Source excerpt

Let $G$ be a graph, $T$ a tree, and let $\mathcal{V} = (V_t)_{t\in T}$ be a family of bags $V_t \subseteq V(G)$ indexed by the nodes $t$ of $T$. The pair $(T,\mathcal{V})$ is called a tree-decomposition of $G$ if it satisfies the following three conditions:

(T1) $V(G) = \bigcup_{t\in T} V_t$;

(T2) for every edge $e \in G$ there exists a $t \in T$ such that both ends of $e$ lie in $V_t$;

(T3) $V_{t_1} \cap V_{t_3} \subseteq V_{t_2}$ whenever $t_1,t_2,t_3 \in T$ satisfy $t_2 \in t_1 T t_3$.

Synthesized QA

Question. Let $G$ be a graph and let $(T,\mathcal{V})$ be a tree decomposition of $G$, where $\mathcal{V} = \{V_t : t \in V(T)\}$ is a family of bags such that $\bigcup_{t\in V(T)} V_t = V(G)$. Given a set of vertices $W \subseteq V(G)$, prove that for any $w \in W$ there exists a node $t' \in V(T)$ with $w \in V_{t'}$.

Answer. By definition of a tree decomposition (condition (T1)), $V(G) = \bigcup_{t\in V(T)} V_t$, i.e. the bags cover all vertices of $G$. Since $W \subseteq V(G)$, every $w \in W$ lies in $V(G) = \bigcup_{t\in V(T)} V_t$, so there exists a node $t' \in V(T)$ with $w \in V_{t'}$. $\square$

L2 – Concept explanation (Section 5.4, List colouring)

Source excerpt

In principle, showing that a given graph is $k$-choosable is more difficult than proving it to be $k$-colourable: the latter is just the special case of the former where all lists are equal to $\{1,\ldots,k\}$. Thus,

for all graphs $G$.

Synthesized QA

Question. Explain, in the setting of list colouring, why $\operatorname{ch}(G) \geq\chi(G)$ and $\operatorname{ch}'(G) \geq\chi'(G)$ for every graph $G$.

Answer. In principle, showing that a given graph is $k$-choosable is more difficult than proving it to be $k$-colourable: the latter is just the special case of the former where all lists are equal to $\{1,\ldots,k\}$. Hence any list of size $k$ can be specialized to this uniform list, and the two inequalities follow.

L3 – Single-concept short proof (Section 1.2, Proposition 1.2.1)

Source excerpt

Summing $d(v)$ over all $v \in V$ counts every edge twice, so

Proposition 1.2.1. The number of vertices of odd degree in a graph is always even.

Proof. As $|E| = \frac{1}{2}\sum_{v\in V} d(v)$ is an integer, $\sum_{v\in V} d(v)$ is even.

Evaluation Protocols

Synthesized QA

Question. Prove that the number of vertices of odd degree in a graph is always even.

Answer. Since $|E| = \frac{1}{2}\sum_{v\in V} d(v)$ is an integer, $\sum_{v\in V} d(v)$ is even. Reducing this sum modulo 2, each even-degree vertex contributes 0 and each odd-degree vertex contributes 1, so $|V_{\mathrm{odd}}| \equiv0 \pmod{2}$; the set of odd-degree vertices has even cardinality. □

L4 – Chapter-local multi-step proof (Section 1.9, Lemma 1.9.3)

Source excerpt

Proposition 1.9.2. Together with $\emptyset$, the cuts in $G$ form a subspace $\mathcal{B}(G)$ of $\mathcal{E}(G)$. This space is generated by atomic cuts. (A minimal non-empty cut is a bond.)

Lemma 1.9.3. Every cut is a disjoint union of bonds.

Proof. We apply induction on the size of the cut $F$ considered. For $F = \emptyset$ the assertion is trivial (with the empty union). If $F \ne\emptyset$ is not itself a bond, it properly contains some other non-empty cut $F'$. By Proposition 1.9.2, $F \setminus F' = F + F'$ is also a smaller non-empty cut. By the induction hypothesis, both $F'$ and $F \setminus F'$ are disjoint unions of bonds, and hence so is $F$. □

Synthesized QA

Question. Prove that every cut in a graph is a disjoint union of bonds.

Answer. We apply induction on the size of the cut $F$. For $F = \emptyset$ the assertion is trivial (with the empty union). If $F \ne\emptyset$ is not itself a bond, it properly contains some other non-empty cut $F'$. By Proposition 1.9.2, $F \setminus F' = F + F'$ is also a smaller non-empty cut. By the induction hypothesis, both $F'$ and $F \setminus F'$ are disjoint unions of bonds, and hence so is $F$. □

L5 – Cross-chapter synthesis (Section 12.7, Corollary 12.7.3)

Source excerpt

Corollary 12.7.3. For every surface $S$ there exists a finite set of graphs $H_1,\ldots,H_n$ such that a graph is embeddable in $S$ if and only if it contains none of $H_1,\ldots,H_n$ as a minor.

While Corollary 12.7.3 is immediate from the graph minor theorem, it can also be proved more directly. The main step is to prove that the graphs in $\mathcal{K}_{P(S)}$ do not contain arbitrarily large grids as minors (Lemma 12.7.4). Then their tree-width is bounded (Theorem 12.6.3), so $\mathcal{K}_{P(S)}$ is well-quasi-ordered (Theorem 12.4.2) and therefore finite.

Synthesized QA

Question. Show that for every surface $S$ there exists a finite set of graphs $H_1,\ldots,H_n$ such that a graph is embeddable in $S$ if and only if it contains none of $H_1,\ldots,H_n$ as a minor.

Answer. By their minimality, the graphs in the Kuratowski set $\mathcal{K}_{P(S)}$ are incomparable under the minor relation. If their tree-width is bounded, then $\mathcal{K}_{P(S)}$ is well-quasi-ordered by the minor relation (Theorem 12.4.2), and hence must be finite. So assume their tree-width is unbounded, and let $r$ be as in Lemma 12.7.4. By Theorem 12.6.3, some $H \in\mathcal{K}_{P(S)}$ has a grid minor large enough to contain $H^r$. By Proposition 1.7.3, $H^r$ is a topological minor of $H$, contrary to the choice of $r$. □

The five examples illustrate the progression of reasoning across the L1–L5 ladder: L1 applies a defining condition of tree-decomposition; L2 explains the logical relationship between list colouring and ordinary colouring; L3 develops a parity argument from the degree-sum identity; L4 combines chapter-local results with induction; and L5 integrates results on grid minors, tree-width, and well-quasi-ordering with an earlier result on topological minors to establish the finite forbidden-minor characterization of surface embeddability.

Evaluation Protocols

Evaluation set construction

TOME-LONGFORM. The long-form component of TOME-EVAL contains 300 textbook-native problems with reference solutions. It covers six subjects, with 25 problems sampled from the first half and 25 from the second half of the chapter sequence in each subject (Table 6). For a problem in the zero-based chapter position $i$ of a book with $N$ chapters, we define the first half by $i/N < 0.5$ and the second half by $i/N \ge0.5$. This 300-problem, six-subject suite is the frozen version used in the completed experiments; it supersedes an earlier 500-problem evaluation snapshot.

Subject	First	Second	Books	Proof	Other	Total
Discrete mathematics	25	25	10	39	11	50
Mathematical analysis	25	25	10	29	21	50
Abstract algebra	25	25	10	46	4	50
Probability theory	25	25	10	28	22	50
Differential geometry	25	25	12	43	7	50
Topology	25	25	10	42	8	50
Total	150	150	62	227	73	300

We freeze source books before constructing any training subset. The held-out manifest contains 64 books: ten per subject except differential geometry, for which fourteen books are held out to provide sufficient coverage of later chapters. All 15,661 training records associated with these books are removed from every training scale, leaving 364,631 eligible records. The final problems come from 62 of the held-out books and 212 distinct book–chapter pairs; the two remaining books stay excluded from training although none of their problems is selected.

Filtering and selection. Starting from 5,058 problems in the held-out books, deterministic filters retain 4,316 candidates. We require a 21–4,000 character question and a 200–8,000 character reference solution, and exclude empty, image-dependent, context-dependent, duplicated, and exact train-overlapping items. A Luna-based screen then annotates reference completeness, self-containment, mathematical relevance, task type, and difficulty. We retain only candidates judged self-contained and mathematical with a usable reference derivation. Within each subject–chapter-half cell, we sample across books, cap each book at six problems, and stratify by the auxiliary difficulty label. The base model is not used to select problems.

All 300 selected items are subsequently reviewed and accepted by a mathematics PhD. They comprise 227 proof, 50 calculation, 18 construction, and five explanation problems. We therefore describe this suite as book-held-out and proof-heavy. We freeze the resulting set before running any model evaluation and use the same version for all comparisons.

TOME-OBJECTIVE. The objective component of TOME-EVAL contains 145 unique problems from 19 subject categories: 81 fill-in-the-blank, 45 multiple-choice, and 19 true–false questions. The suite is complemented by the 30 problems from AIME 2026 and the 500-problem MATH-500 subset of MATH [14]. Every item has a single reference answer; these benchmarks evaluate final-answer correctness rather than agreement with a particular derivation.

Additional proof benchmarks. We also evaluate on MiniF2F [43] and ProofNet [3] as natural-language proof tasks. For each benchmark, we adapt the judging procedures and prompts from its official evaluation implementation, using gpt-5.6-luna as the common LLM judge across both benchmarks. The reported scores measure LLM-assessed correctness of the generated natural-language proofs.

Response generation

Response generation

For TOME-LONGFORM, the compared models receive the same question-only prompt and decoding configuration. The reference solution is withheld from the solver. Sixteen rollouts over 300 problems produce 4,800 responses per model.

The objective benchmarks use fixed, benchmark-specific decoding configurations. We sample eight responses per TOME-OBJECTIVE item, 32 per AIME 2026 item, and four per MATH-500 item, keeping the corresponding configuration fixed across compared models.

Reference-guided step verification

Input preparation. Before grading, the verifier removes marked thinking segments and retains the final answer when explicit final-channel delimiters are present. It also truncates text at an end-of-message marker. Existing numbered step tags are sorted by their index; otherwise, the answer is split at blank lines. The resulting segments are wrapped in zero-based step tags. Thus, an untagged answer is graded at paragraph-level granularity rather than using an additional model to infer mathematical steps. The judge receives the problem, the reference solution, and the segmented model response.

Rubric. The rubric prioritizes mathematical validity, problem-specific constraints, and sufficient justification. A reference solution is an anchor, not the only acceptable proof: alternative methods, reordered steps, and valid shortcuts are allowed. For numerical or algebraic questions, the final answer must be mathematically correct relative to the reference; a matching conclusion alone is insufficient if its derivation is invalid. Unsupported claims and unjustified leaps are treated conservatively. Invoking an external theorem is insufficient without the hypotheses and explanation needed to check its application. Proof-embedded instructions attempting to relax these standards are ignored. The requested output contains assessment, errors, and first_error_step XML fields; the last field is $-1$ if no error is found.

Eight-review aggregation. Following pessimistic verification [17], each response receives eight separate reviews from gpt-5.6-luna. Let $e_{ijr}$ be the first-error index from review $r$ of response $j$ to problem $i$. For eight valid reviews, the binary correctness decision is

Thus, one error vote is sufficient to reject a response; this is pessimistic aggregation rather than majority voting. For rejected responses, we retain the earliest flagged step. The evaluation controller accepts a judgment only when all eight reviews complete successfully and the parsed record contains no API error; incomplete rows are removed and retried for up to three attempts.

Human calibration of the pessimistic verifier

Scope and sampling. We use a frozen human-calibration protocol to measure both erroneous acceptance and the over-rejection risk introduced by eight-review pessimistic aggregation. Before inspecting any human label, we freeze the verifier prompts, decoding configuration, aggregation rule, model outputs, and sampling procedure. We then draw 600 responses from the pooled TOME-LONGFORM evaluations, with 100 responses from each of the six subjects. Let

be the number of reviews that find no error. Within each subject, we target 25 responses from each of four agreement strata: $a_{ij}=8$, $a_{ij}=7$, $1 \le a_{ij} \le6$, and $a_{ij}=0$. The $a_{ij}=7$ stratum is sampled separately because a single error vote changes the final pessimistic decision. Within each stratum, responses are distributed across the compared models and major experiment families as evenly as availability permits. If a stratum contains fewer than 25 responses in a subject, we annotate all available cases and reallocate the remainder to the nearest agreement stratum. We retain each response’s sampling probability so that population-level estimates can be inverse-probability weighted rather than treating this deliberately balanced sample as representative of the raw response distribution.

Blinding and annotation materials. Annotators are mathematics PhD holders, or comparably qualified researchers, with expertise appropriate to the assigned subject. For each response, an annotator receives the problem, the reference solution, and the same segmented, step-indexed model response supplied to the automatic verifier. Model identity, training condition, automatic judgments, review agreement, and aggregate score are hidden. The reference is treated as an anchor rather than a unique target: alternative proof methods, reordered arguments, and valid shortcuts are accepted when they establish the requested result. This response-level audit is separate from the earlier human review of the evaluation questions and reference solutions.

Human labels. Each response receives exactly one of the following labels.

• Fully correct: the conclusion and supporting argument are mathematically valid and sufficiently justified.

• Correct with a non-substantive issue: the response contains only a stylistic, notational, typographical, or presentation defect that does not affect mathematical validity.

• Substantive error: the response contains a false statement, invalid inference, missing required hypothesis, circular argument, incorrect calculation, materially incomplete proof, or incorrect final answer.

• Unjudgeable: ambiguity or a defect in the problem, reference, or response prevents a reliable correctness decision.

The first two labels map to human acceptance and Substantive error maps to rejection. Unjudgeable cases require adjudication and are not mapped to a binary label beforehand. For every rejected response, the annotator records the zero-based index of the first substantive error and selects an error category: false claim, invalid inference, missing hypothesis, theorem misuse, incomplete proof, algebraic or numerical error, final-answer error, or other. Every annotation also includes a short justification and a high, medium, or low confidence rating. Purely stylistic differences and harmless notation changes must not trigger rejection.

Independent review and adjudication. One qualified annotator labels all 600 responses. A second qualified annotator independently labels a stratified 30% subset (180 responses), preserving the subject and agreement-stratum proportions and remaining blind to the first annotation. All binary disagreements, all Unjudgeable labels, and all low-confidence cases are adjudicated by a third domain-qualified annotator. If the adjudicator finds the evaluation item itself invalid, that problem is removed for every model and all affected aggregate scores are recomputed.

Calibration statistics. Let $h_{ij}=1$ denote the adjudicated human acceptance label and let $y_{ij}$ be the pessimistic verifier decision in Equation (1). Using the recorded sampling weights, we report the false-acceptance rate $\Pr(h_{ij}=0 \mid y_{ij}=1)$, the false-rejection rate $\Pr(h_{ij}=1 \mid y_{ij}=0)$, overall agreement, and the human acceptance rate within each value of $a_{ij}$. We additionally report raw agreement and Cohen’s $\kappa$ between the two human annotators, together with exact and within-one-step agreement for the first-error index when both reject. All quantities are reported overall and by subject with 95% bootstrap confidence intervals clustered by problem. Finally, we recompute model-level correctness on the weighted human-labeled subset to test whether the principal model ranking and pairwise conclusions agree with the automatic evaluation.

Table 8: Human calibration of the eight-review pessimistic verifier. All rate entries are percentages and should be reported as the weighted point estimate followed by the problem-clustered 95% bootstrap confidence interval in brackets. H–L denotes human–Luna agreement; H–H denotes agreement between the two human annotators.

Panel A: Calibration by subject
Group	Annotated $n$	Human accept	False accept	False reject	H–L agree
Abstract algebra	100	42.1 [35.8, 48.7]	9.8 [4.2, 17.6]	19.3 [13.1, 26.5]	84.7 [79.6, 89.2]
Differential geometry	100	37.8 [31.5, 44.4]	11.4 [5.6, 19.2]	20.7 [14.5, 28.0]	83.5 [78.3, 88.1]
Discrete mathematics	100	52.3 [45.8, 58.7]	7.2 [2.6, 14.0]	16.9 [11.2, 23.7]	87.4 [82.6, 91.5]
Mathematical analysis	100	54.6 [48.1, 60.9]	6.8 [2.4, 13.4]	15.8 [10.3, 22.4]	88.1 [83.5, 92.0]
Probability	100	55.4 [48.9, 61.6]	7.9 [3.0, 15.0]	17.5 [11.7, 24.3]	86.6 [81.7, 90.8]
Topology	100	45.2 [38.7, 51.8]	9.1 [3.8, 16.8]	18.6 [12.5, 25.7]	85.2 [80.2, 89.6]
Overall	600	47.9 [44.6, 51.2]	8.7 [6.3, 11.5]	18.1 [15.3, 21.1]	85.9 [83.8, 87.9]
Panel B: Calibration by number of no-error votes
Luna votes $a_{ij}$	Annotated $n$	Population share	Human accept	Human substantive error	H–L agree
8 of 8	150	42.3 [40.9, 43.7]	91.3 [86.4, 94.8]	7.5 [4.1, 12.3]	91.3 [86.4, 94.8]
7 of 8	150	7.8 [7.0, 8.7]	68.4 [60.5, 75.6]	30.2 [23.1, 38.2]	31.6 [24.4, 39.5]
1–6 of 8	150	18.1 [17.0, 19.3]	22.1 [15.9, 29.4]	76.4 [69.0, 82.7]	77.9 [70.6, 84.1]
0 of 8	150	31.8 [30.5, 33.1]	2.8 [0.9, 6.4]	96.4 [92.3, 98.6]	97.2 [93.6, 99.1]
Panel C: Human–human reliability on the double-coded subset
Subset	Double-coded $n$	H–H agree	Cohen’s $\kappa$	Exact error step	Within one step
Stratified 30% subset	180	89.4 [84.9, 93.1]	0.771 [0.687, 0.842]	52.8 [43.2, 62.1]	76.9 [68.5, 84.1]

Objective and additional proof scoring

For TOME-OBJECTIVE, a low-temperature Qwen2.5-32B-Instruct judge compares the extracted final answer with the reference. It accepts mathematically equivalent expressions and requires every component of a multipart or multiple-selection answer to be correct. The judge receives the problem, reference answer, and prediction, but not the worked reference solution. AIME 2026 and MATH-500 first use the rule-based MATHVerifyEvaluator; cases not resolved by the rule-based parser fall back to the same low-temperature LLM equivalence judge. These evaluators score answer correctness only and do not assess the generated reasoning path. For MiniF2F and ProofNet, we assess natural-language proof correctness using judging procedures and prompts adapted from their official evaluation implementations, with gpt-5.6-luna as the common LLM judge.

Metrics and validity checks

For a problem with $n$ sampled responses, of which $c_i$ are correct, we use the standard unbiased estimator [5]

For TOME-LONGFORM, $n=16$ and we report pass@1, pass@2, pass@4, pass@8, and pass@16, with pass@1 and pass@16 as the primary metrics. Because $n=k=16$, pass@16 is also the observed fraction of problems solved at least once. We report per-subject scores and their macro average; the equal 50-problem subject sizes make this macro average equal to the overall problem average. For TOME-OBJECTIVE, the mean correctness across its eight samples estimates pass@1; pass@$k$ is computed separately when reported.

For every reported model, we require the complete expected number of rollouts for every problem. For natural-language proof judgments, we additionally require eight successful reviews per response and rejudge any incomplete row. We preserve raw judge outputs and parsed first-error indices for audit.

Full-Corpus Comparison: Extended Results

Full-Corpus Comparison: Extended Results

This appendix expands § 4.2 (“Full-Corpus Comparison”) with uncertainty estimates and per-corpus training configurations that were omitted from the main text for space. All fine-tuned models share the Qwen3-30B-A3B-Base initialization and the evaluation protocol described in Appendix § 11; the compared corpora differ only in their supervision data.

Compared corpora

Public SFT corpora. We use the official Hugging Face releases of three public corpora. OpenMathInstruct-1 (abbreviated OpenMath1 in our tables) comes from nvidia/OpenMathInstruct-1 at revision 4627efae. MetaMathQA comes from meta-math/MetaMathQA at revision aa4f34d3, and MathInstruct comes from TIGER-Lab/MathInstruct at revision b4fdc323 [32, 39, 40]. OpenMathInstruct-1 is distributed under the NVIDIA Dataset License and MetaMathQA is labeled MIT; MathInstruct combines source datasets under their respective licenses, including MIT, Apache-2.0, and CC BY-NC 4.0 components. We retain the released training examples (the correct-solution subset for OpenMathInstruct-1), map their native question and response fields to the same user–assistant training template, and apply no additional cross-corpus deduplication.

TOME-Math (ours). We use the frozen full-corpus snapshot produced by the pipeline in Appendix § 7: 380,292 records before evaluation holdout filtering. Removing all 15,661 records from held-out evaluation books leaves 364,631 training examples. We retain all remaining quality tiers and the corpus’s natural subject distribution, without subject rebalancing or further deduplication.

Corpus	Training examples	Training tokens	Notes
OpenMath1	1.82M	368.00M	Full corpus
MetaMath	395k	96.94M	Full corpus
MathInstruct	262k	64.10M	Full corpus
TOME-Math (ours)	364,631	170.92M	Held-out books excluded

Training configuration

Each model is trained for exactly one full epoch over its respective training corpus. We use the same optimizer, learning-rate schedule, sequence length, effective batch size, and data format across corpora. The complete TOME-Math release contains 380,292 examples and 195.18M tokens; excluding the 15,661 records associated with the held-out evaluation books leaves 364,631 training examples and approximately 170.92M training tokens.

Setting	Value
Base model	Qwen3-30B-A3B-Base
Optimizer	AdamW ($\beta_{1}=0.9$, $\beta_{2}=0.95$; weight decay = 0.1)
Learning rate	$8 \times10^{-5}$, decayed to a minimum of $8 \times10^{-6}$
Warmup / schedule	3% warmup followed by cosine decay (0 warmup steps in the 18-step run)
Sequence length	131,072 tokens; global packing
Effective batch size	64 (micro-batch size 1/GPU; gradient accumulation 4)
Epochs	1
Hardware	$16\times$ NVIDIA H200 (2 nodes $\times$ 8 GPUs)

All benchmark results

For completeness, Table 11 reproduces the six-benchmark comparison from Table 2. We estimate pass@1 from 16 responses per problem for TOME-LF, MiniF2F, and ProofNet, eight for TOME-OBJECTIVE, four for MATH-500, and 32 for AIME 2026. For MiniF2F and ProofNet, the natural-language proofs are assessed using judging procedures and prompts adapted from their official evaluation implementations, with gpt-5.6-luna as the common LLM judge. We compute 95% problem-clustered bootstrap confidence intervals using 2,000 resamples. AIME 2026 has the widest intervals because it contains only 30 problems. The remaining scoring details follow Appendix § 11.

Model (training data)	TOME-LF	TOME-Obj.	MiniF2F	ProofNet	MATH-500	AIME-26
Qwen3-30B-A3B-Base	$13.25 \pm3.35$	$32.93 \pm8.20$	$34.45 \pm6.00$	$17.14 \pm3.95$	$68.80 \pm4.40$	$8.44 \pm8.10$
OpenMath1 (1.82M)	$20.16 \pm4.10$	$33.86 \pm8.30$	$22.87 \pm5.40$	$19.25 \pm4.15$	$70.40 \pm4.30$	$8.21 \pm8.05$
MetaMath (395k)	$28.73 \pm4.85$	$34.66 \pm8.35$	$40.70 \pm6.15$	$22.55 \pm4.40$	$71.20 \pm4.30$	$9.98 \pm8.70$
MathInstruct (262k)	$35.87 \pm5.20$	$40.69 \pm8.65$	$52.05 \pm6.20$	$32.93 \pm5.25$	$73.05 \pm4.20$	$11.98 \pm9.10$
TOME-Math (ours; 364.6k)	$43.86 \pm5.40$	$43.96 \pm8.75$	$57.43 \pm6.10$	$42.98 \pm5.45$	$76.55 \pm3.90$	$10.19 \pm8.75$

Training-Free Dependency-Context Study

Evaluation set. The study in § 5.1 uses the 300-problem TOME-LONGFORM evaluation set described in Appendix § 11. It contains 50 textbook-native problems from each of abstract algebra, differential geometry, discrete mathematics, mathematical analysis, probability, and topology. The source books are excluded from the TOME-Math training corpus. Every problem is evaluated under all five context conditions, making the comparison paired at the problem level.

Retrieval inventories. Candidate context is drawn only from TOME-Math records associated with the exercise’s source book. The proposition inventory contains extracted definitions, lemmas, theorems, propositions, corollaries, and their associated proof records. The expanded inventory adds the self-contained local proof obligations and answers from the Gap QA stream. Source type, chapter position, and recovered dependency metadata are retained for retrieval.

The five conditions are constructed as follows.

• CLOSED: provide only the exercise.

• RANDOM SAME-BOOK: sample entries uniformly from the same source book, without using chapter or dependency information.

• LOCAL CHAPTER: restrict candidates to the exercise chapter and its immediately preceding chapters, adding locality while ignoring recovered cross-chapter dependencies.

• TOME (PROPS ONLY): retrieve from the proposition inventory using chapter position and the recovered proposition dependencies.

• TOME (PROPS + GAP): use the same structure-guided retrieval procedure over the union of the proposition and Gap QA inventories.

Context and leakage controls. All four context-augmented conditions use the same context-token budget and the same prompt template; only the candidate pool and retrieval constraint change. Retrieved entries are placed before the exercise and identified by their source type. The reference solution is never shown to the solver. Candidates that reproduce the target exercise or overlap with its reference answer under the frozen leakage filter are removed before retrieval. The same filtered inventory is used to construct the random and local baselines, so the comparison does not give the TOME conditions access to otherwise unavailable source material.

Generation and scoring. The solver is the pretrained Qwen3-30B-A3B-Base in every condition. Each problem receives 16 sampled responses under the decoding configuration in Table 7; adding context does not change the model weights, prompt instruction, or sampling parameters. Responses are graded with the same reference-guided eight-review verifier used for TOME-LONGFORM. We compute pass@1 as the mean binary correctness across the 16 responses for each problem, average within each of the six equally sized subjects, and report their unweighted mean in the Overall column of Table 3.

Context	Avg.	Abs.	Alg.	Diff.	Geom.	Discrete	Math.	Anal.	Prob.	Topology
CLOSED	13.25	8.50	5.25	13.25	25.75	20.00	6.75
RANDOM SAME-BOOK	14.00	9.75	6.50	12.50	24.75	20.50	10.00
LOCAL CHAPTER	14.71	10.50	6.75	14.50	25.50	20.00	11.00
TOME (PROPS ONLY)	15.54	11.00	8.50	15.00	27.25	20.50	11.00
TOME (PROPS + GAP)	17.38	12.75	10.00	15.25	31.75	21.75	12.75

Setting	Value
Responses per question	16
Temperature	0.6
Top-$p$	0.8
Maximum output tokens	16,384
Completions per request	1
Presence / frequency penalty	0 / 0

Result interpretation. The aggregate score increases from 13.25% without context to 14.00% with random same-book context and 14.71% with local context. Proposition-based TOME retrieval reaches 15.54%, and adding the Gap QA inventory raises the score to 17.38%. The final condition is best in every subject, although the intermediate conditions are not monotonically ordered within every subject. Accordingly, the result supports the complete structure-guided retrieval design and the complementary value of Gap QA.

Curriculum-Scaling Protocol

Subject-wise design. The curriculum study in § 5.2 covers discrete mathematics, mathematical analysis, abstract algebra, probability, differential geometry, and topology. We train a separate model for each subject so that curriculum order is not confounded with the composition of a multidisciplinary corpus. Every condition is evaluated on the same textbook-native exercises from held-out books in the corresponding subject.

Normalized chapter stages. Textbooks contain different numbers of chapters, so raw chapter indices are not directly comparable across books. For an example $x$ from book $b$, let $i(x)$ be its zero-based chapter index and $N_b$ the number of chapters in the book. We define its normalized position as $p(x) = i(x)/N_b$ and assign it to progress stage $S_{\lfloor10p(x)\rfloor}$, yielding ten stages $S_0,\ldots,S_9$. Examples from different books are interleaved within each stage, preserving their approximate within-book progression without imposing an arbitrary order over books. Gap QA makes a locally omitted proof obligation explicit; all remaining examples are treated as Proof QA.

Exposure schedules. The four schedules differ only in presentation order:

• SHUFFLE globally shuffles all examples.

• FORWARD–GAP traverses stages from $S_0$ to $S_9$ and presents Gap before Non-gap examples within each stage.

• REVERSE–GAP traverses stages from $S_9$ to $S_0$ while retaining the same Gap-first policy.

• FORWARD–MIX follows the forward stage order but mixes Gap and Non-gap examples within each stage.

Thus, FORWARD–GAP versus REVERSE–GAP is the controlled comparison for chapter direction, and FORWARD–GAP versus FORWARD–MIX is the controlled comparison for stage-wise Gap-first ordering. The comparison with SHUFFLE changes both factors and is interpreted as the combined effect of the structured schedule.

Scales and matched controls. We use per-subject target-token budgets of 0.75M, 1.8M, and 2.4M. Within every subject, both the source-book sets and the QA sets are strictly nested across scales: the 8-book set is contained in the 16-book set, which is contained in the 24-book set, and the corresponding QA sets follow the same nesting. Thus, the three columns in Table 12 are cumulative rather than disjoint. All held-out evaluation books are excluded from every training scale.

Target tokens, rather than the number of books or QA pairs, define supervision scale because answer lengths vary across sources and subjects. The realized token totals across the six subjects are 4,499,926, 10,799,935, and 14,399,975 for the three scales, respectively; each subject-level total differs from its nominal target by at most 33 tokens. Within each subject–scale cell, all schedules use identical QA IDs, source books, realized token counts, model initialization, and optimization hyperparameters. Data-loader shuffling, sequence packing, and length-based grouping are disabled so that the prescribed exposure order is preserved.

Matched-budget source breadth. The source-breadth comparison in § 5.3 reuses the 8-book, 0.75M-token FORWARD–GAP condition as NARROW–DENSE. For BROAD–SPARSE, we take the corresponding nested 16-book pool and downsample its QA records to the realized token count of the 8-book condition, while retaining examples from all 16 books. The matched pair uses the same optimization steps, schedule construction, initialization, and held-out-book evaluation set. Consequently, this comparison doubles source coverage without increasing the training-token budget; it is distinct from comparing the 0.75M- and 1.8M-token checkpoints in the scale study. Scores in Figure 5(a) are averaged over five pass@16 evaluation runs per checkpoint, with 16 responses sampled per problem in each run.

Cross-Subject Transfer Protocol and Results

Checkpoint selection. The study in § 5.3 uses one 0.75M-token FORWARD–GAP checkpoint per subject from § 5.2. Each checkpoint is trained exclusively on data from its corresponding subject; initialization, optimization, and schedule construction are shared across subjects.

Evaluation sets. We evaluate each checkpoint on 50 textbook-native held-out-book exercises in each of the other five subjects. Drawing 16 responses per exercise with the configuration in Table 7 yields 4,000 responses per checkpoint and 24,000 overall. The within-subject diagonal is not evaluated; the corresponding results appear in § 5.2.

Aggregation. For each ordered source–target pair, pass@16 is computed over 50 exercises. Source and target means average their five off-diagonal scores, and the macro mean weights all 30 pairs equally. Table 13 reports the matrix underlying Figure 5(b). Table 14 reports the complete cross-subject pass@$k$ results for $k \in\{1, 2, 4, 8, 16\}$.

Table 13: Exact cross-subject pass@16 scores (%) for the 0.75M-token FORWARD–GAP checkpoints. Rows denote training subjects and columns denote held-out evaluation subjects. The diagonal was not evaluated; marginal means exclude it.

Train \ Eval	Alg.	Ana.	Prob.	DG	Disc.	Top.	Mean
Alg.	–	60.0	52.0	26.0	40.0	26.0	40.8
Ana.	28.0	–	46.0	32.0	40.0	36.0	36.4
Prob.	34.0	66.0	–	24.0	42.0	38.0	40.8
DG	36.0	62.0	54.0	–	44.0	34.0	46.0
Disc.	30.0	56.0	48.0	30.0	–	38.0	40.4
Top.	34.0	62.0	54.0	32.0	38.0	–	44.0
Mean	32.4	61.2	50.8	28.8	40.8	34.4	41.4

Table 14: Complete cross-subject pass@$k$ scores (%) for the six 0.75M-token FORWARD–GAP checkpoints. Each block fixes the training subject and reports performance on the other five held-out subject sets. Every target set contains 50 exercises with 16 sampled responses per exercise.

Train	Eval.	pass@1	pass@2	pass@4	pass@8	pass@16
Trained on Abstract Algebra
Alg.	Ana.	26.88	36.92	45.42	53.05	60.00
Alg.	Prob.	20.00	27.95	36.09	43.80	52.00
Alg.	DG	8.13	12.03	16.47	20.82	26.00
Alg.	Disc.	15.88	21.50	27.38	33.20	40.00
Alg.	Top.	10.00	15.25	20.06	23.39	26.00
Trained on Mathematical Analysis
Ana.	Alg.	11.00	15.80	20.10	24.15	28.00
Ana.	Prob.	20.13	27.95	34.63	40.43	46.00
Ana.	DG	8.50	13.13	19.05	25.75	32.00
Ana.	Disc.	17.38	22.62	28.30	34.17	40.00
Ana.	Top.	10.25	16.35	23.92	31.13	36.00
Trained on Probability Theory
Prob.	Alg.	10.50	16.23	22.37	28.38	34.00
Prob.	Ana.	26.25	34.98	44.34	54.38	66.00
Prob.	DG	10.13	14.58	18.73	22.07	24.00
Prob.	Disc.	16.88	23.17	29.20	35.53	42.00
Prob.	Top.	12.63	19.38	26.09	32.22	38.00
Trained on Differential Geometry
DG	Alg.	13.25	19.22	25.08	30.56	36.00
DG	Ana.	29.13	38.42	46.22	54.16	62.00
DG	Prob.	24.63	34.03	42.09	48.15	54.00
DG	Disc.	20.88	26.83	32.84	38.46	44.00
DG	Top.	10.88	16.43	22.55	28.94	34.00
Trained on Discrete Mathematics
Disc.	Alg.	9.38	14.85	20.60	25.46	30.00
Disc.	Ana.	23.63	32.52	40.80	48.50	56.00
Disc.	Prob.	19.50	27.25	34.25	40.73	48.00
Disc.	DG	7.63	11.57	16.50	22.85	30.00
Disc.	Top.	10.88	16.90	23.61	30.99	38.00
Trained on Topology
Top.	Alg.	11.88	17.53	23.35	28.54	34.00
Top.	Ana.	26.38	36.17	45.75	54.90	62.00
Top.	Prob.	23.25	31.08	38.55	46.46	54.00
Top.	DG	7.38	11.50	17.02	23.83	32.00
Top.	Disc.	16.00	21.02	26.21	32.31	38.00