# Data schema and provenance

The machine-readable data are a small research index, not a measured dataset about human behavior, a trained model, or a validated machine-learning benchmark. There are two JSON Lines files with one object per line. The original images are not included.

## `claims`

Every theorem, proposition, lemma, corollary, and conjecture environment in the supplied manuscript is extracted once, in source order. Definitions, unnumbered calculations, audit prose, and open questions outside those environments are not separate rows. Proofs are copied without rewriting; source macros and cross-references are retained. This deliberate extraction boundary prevents treating the index as the full paper.

| Field | Type | Meaning |
|---|---|---|
| `claim_id` | string | Original unique LaTeX label. |
| `title` | string | Original environment title. |
| `environment` | string | `theorem`, `proposition`, `lemma`, `corollary`, or `conjecture`. |
| `component` | string | Main area-feedback construction, algebraic alternative, or spectral appendix. |
| `status` | string | `manuscript_proved` or `conjecture`. |
| `statement_latex` | string | Complete environment body, including its label. |
| `proof_latex` | string | Immediately associated manuscript proof, or empty for the conjecture. |
| `references_in_statement_or_proof` | list of strings | Syntactic `ref` and `eqref` targets, not a certified logical dependency graph. |
| `source_file` | string | Repository-relative standalone LaTeX path. |
| `proof_audit_file` | string | Repository-relative original audit path. |
| `priority_status` | string | Priority is not certified at the record level. |
| `verification_limit` | string | Meaning and limits of the manuscript proof label. |

`manuscript_proved` means a proof is supplied in the paper. It does not assert acceptance by a journal, a human expert, or a proof assistant. Many constituent ideas are established; the index makes no novelty claim for every row.

## `worked_examples`

Eight curated finite calculations are serialized using exact rational numbers. Matrices and control lists are stored as JSON **strings** so all dimensions, including zero, share one simple tabular schema. Parse them with `json.loads` and exact fractions, never by `eval`.

| Field | Type | Meaning |
|---|---|---|
| `example_id` | string | Stable descriptive ID. |
| `dimension` | integer | Ambient dimension. |
| `initial_state_json` | string | JSON object with `x` and `A`, with rational-string entries. |
| `controls_json` | string | JSON list of displacement vectors with rational-string entries. |
| `all_controls_legal` | boolean | Whether every proposed input is legal. |
| `first_illegal_step` | integer | One-based index of the first illegal input; zero if none. |
| `final_legal_prefix_state_json` | string | State after the legal prefix only. |
| `final_rank` | integer | Rank of the final legal-prefix operator. |
| `final_rho_squared` | string | Exact rational squared projection radius. |
| `note` | string | Calculation context and interpretation. |
| `source_file` | string | Manuscript containing the underlying construction and examples. |

Two examples use the same finite controls with different explanatory roles; this is not an independent sampling claim. The illegal rectangle record intentionally stops before its illegal input. It does not apply an illegal update to reach its reported state.

## Splits and appropriate use

Each configuration uses the conventional `train` split solely for simple Hub loading. There is no training/validation/test partition. These examples are deliberately curated, small, and dependent on the defining construction. They should not be used to claim out-of-distribution reasoning performance, unbiased coverage, or statistical superiority.

Natural uses are claim retrieval, proof auditing, exact-example reproduction, comparison with related mathematics, and extension experiments. If used for model training or evaluation, first resolve reuse permission and record this narrow provenance. See [the license notice](../LICENSE_NOTICE.md).