Add answers config (154 verified answers) and math_intent labels on numerical

README.md

@@ -8,7 +8,7 @@ tags:
 - mathematics
 - arxiv
 - open-problems
-- benchmark-format
+- research-level
 size_categories:
 - 10K<n<100K
 configs:
@@ -20,153 +20,146 @@ configs:
   data_files:
   - split: train
     path: data/yesno-*.parquet
+- config_name: answers
+  data_files:
+  - split: train
+    path: data/answers-*.parquet
 ---
 
 # ResearchMath-2-6Sol-Rewrite
 
-**97,497 research-level mathematics questions reformatted into benchmark-shaped items**,
-derived from [`amphora/ArXivOpenProblems`](https://huggingface.co/datasets/amphora/ArXivOpenProblems).
-
-Two configs:
+**97,497 research-level mathematics questions** reformatted into benchmark-shaped items, derived
+from [`amphora/ArXivOpenProblems`](https://huggingface.co/datasets/amphora/ArXivOpenProblems),
+plus an **answer key of 154 literature-verified answers**.
 
-| config | rows | the answer is |
+| config | rows | contents |
 |---|---:|---|
-| `numerical` | **35,274** | a single number |
-| `yesno` | **62,223** | Yes or No |
+| `numerical` | **35,274** | answer is a single number; labelled with `math_intent` |
+| `yesno` | **62,223** | answer is Yes or No |
+| `answers` | **270** (154 verified) | literature search results, with link, theorem, derivation and solution sketch |
 
-## ⚠️ There are no answer keys
+## ⚠️ Most items have no answer key
 
-Every item derives from a **future-work or open-problem statement** in an arXiv paper. The
-answers are, by construction, **not known** — not to the source paper and in most cases not to
-the literature. This is a corpus of *well-posed unanswered questions*, **not a scoreable
-benchmark**. Do not use it to compute accuracy without sourcing answers independently.
+Every item derives from a **future-work or open-problem statement** in an arXiv paper, so for the
+vast majority the answer is **not known**. The `answers` config covers only the small subset where
+a later paper resolved the question — see below. Do not compute accuracy on `numerical`/`yesno`
+without an answer source.
 
 ## How a research question becomes a number
 
-Most source questions ask whether something holds, not for a value. The conversion uses seven
-routes; `route` records which was applied.
+`route` records the construction used for numerical items:
 
-| route | construction | share of `numerical` |
+| route | construction | share |
 |---|---|---:|
 | **B** | minimal counterexample: "smallest *p* at which the property fails; −1 if none" | **80.8%** |
 | E | size of the exceptional set (0 = the original's "yes") | 6.6% |
 | A | the question already asked for a quantity | 6.0% |
-| C | the sharp constant in an inequality | 2.5% |
-| D | the critical value of a continuous parameter | 2.4% |
+| C | sharp constant in an inequality | 2.5% |
+| D | critical value of a continuous parameter | 2.4% |
 | G | count of a finite classification | 1.4% |
-| F | minimal/maximal multiplicity or order (1 = the original's "yes") | 0.3% |
+| F | minimal/maximal multiplicity or order | 0.3% |
 
-Route B dominates. `distinguished_value` gives the number corresponding to the original
-question's "yes" — usually `-1`.
+`distinguished_value` is the number equal to the original question's "yes" (usually `-1`).
+**Route B strengthens the problem**: "what is the smallest counterexample?" is strictly harder than
+"is there one?".
 
-**Route B strengthens the problem.** "Is there a counterexample?" and "what is the smallest
-counterexample?" are different questions; the second is strictly harder. The reformulation is
-faithful in that the distinguished value is equivalent to the original's affirmative answer,
-but the item is not of equal difficulty to its source.
+## `math_intent` (numerical config)
 
-## Fields
+What each question is *about* — orthogonal to `question_category` (its logical shape). Assigned by
+`openai/gpt-6-sol` from the source question; every one of the 35,274 rows is labelled.
 
-| field | description |
-|---|---|
-| `uid` | `<arxiv_id>#<index>`, joins to the source dataset |
-| `arxiv_id`, `paper_url` | provenance |
-| `primary_category` | arXiv category of the source paper |
-| `signal_type` | source statement type: `conjecture`, `limitation`, `open_problem`, `natural_extension`, `announced_forthcoming` |
-| `question_category` | logical shape of the **original** question (see below) |
-| `format` | `numerical` or `yesno` |
-| `route` | `A`–`G` for numerical items, `null` for yes/no |
-| `question` | the reformatted item |
-| `answer_convention` | one sentence on what the answer means |
-| `distinguished_value` | number equal to the original's "yes", or `null` |
-| `source_question` | the unmodified original |
-| `engine` | `openai/gpt-6-sol` |
-| `math_intent` | **numerical config only.** What the source question is *about* — see [Mathematical intent](#mathematical-intent) |
-| `secondary_intent` | **numerical config only.** A second applicable intent, or `null` |
-
-`question_category` is one of `universality`, `existence`, `extension`, `characterization`,
-`construction`, `comparison`, `classification`, `quantity` — assigned during conversion, so no
-re-classification is needed.
-
-## What converts, and what doesn't
-
-Numerical yield by logical shape of the source question:
-
-| question_category | → numerical |
-|---|---:|
-| universality ("does P hold for every X") | **~65%** |
-| quantity | ~59% |
-| classification | ~42% |
-| existence | ~29% |
-| extension | ~21% |
-| construction | ~19% |
-| characterization | ~13% |
-| comparison | ~9% |
-
-By source statement type, measured on the full run:
-
-| signal_type | n | → numerical |
+| math_intent | n | share |
 |---|---:|---:|
-| conjecture | 39,444 | **50%** |
-| open_problem | 8,756 | 37% |
-| announced_forthcoming | 897 | 31% |
-| limitation | 41,265 | 25% |
-| natural_extension | 7,197 | 24% |
-
-Universality statements convert because negating "for all *n*" yields a minimal counterexample.
-Questions asking for a *construction*, a *characterization*, or a qualitative *comparison* have
-no number in them, and no reformulation produces one — those became yes/no items.
-
-## Mathematical intent
-
-The `numerical` config carries a second classification axis. `question_category` records the
-**logical shape** of the source question (universality, existence, …); `math_intent` records what
-it is **about**. The two are largely independent — bound-improvement questions, for instance, are
-spread across most logical shapes.
-
-| math_intent | count | share | the source question asks about… |
-|---|---:|---:|---|
-| `structural_characterization` | 7,423 | 21.0% | characterise or describe a structure, invariant or behaviour |
-| `existence_construction` | 6,802 | 19.3% | whether an object exists, or construct / exhibit one |
-| `generalization` | 6,749 | 19.1% | whether a result extends to a larger setting (higher dimension, general fields, …) |
-| `bound_improvement` | 6,374 | 18.1% | improve / sharpen a bound, constant, exponent or rate, or ask whether one is optimal |
-| `hypothesis_weakening` | 2,545 | 7.2% | whether a hypothesis or regularity assumption can be weakened or removed |
-| `classification` | 1,590 | 4.5% | classify or enumerate all objects with a property |
-| `asymptotic_rate` | 1,535 | 4.4% | an asymptotic, growth rate or order of magnitude |
-| `uniqueness` | 971 | 2.8% | uniqueness, or whether exactly one object has a property |
-| `algorithmic` | 677 | 1.9% | algorithms, computability, decidability or complexity |
-| `other` | 608 | 1.7% | none of the above |
-
-`secondary_intent` is set for 13,365 rows (38%) where a second label clearly applies.
-
-Labels were assigned by `openai/gpt-6-sol` reading the **original** source question (the
-reformatted question often obscures intent — a minimal-counterexample rewrite no longer reads as a
-bound question). Only 1.7% fell into `other`. These are model labels, not expert annotation; the
-`yesno` config is not yet labelled.
-
-## Construction and filtering
-
-Converted with `openai/gpt-6-sol` via the OpenRouter batch API, temperature 0.2. 98,000 source
-questions in, 97,497 kept. Removed: 441 unparseable responses, 55 items where a yes/no answer
-had been encoded numerically (indicator variables, `2a+b`), 4 degenerate short items, 3
-duplicate uids.
-
-The conversion prompt forbade encoding booleans as numbers; the residual rate was 0.16% before
-filtering and 0 after.
+| structural_characterization | 7,423 | 21.0% |
+| existence_construction | 6,802 | 19.3% |
+| generalization | 6,749 | 19.1% |
+| **bound_improvement** | **6,374** | **18.1%** |
+| hypothesis_weakening | 2,545 | 7.2% |
+| classification | 1,590 | 4.5% |
+| asymptotic_rate | 1,535 | 4.4% |
+| uniqueness | 971 | 2.8% |
+| algorithmic | 677 | 1.9% |
+| other | 608 | 1.7% |
+
+`math_intent_secondary` gives a second label where one clearly applies.
+
+## The `answers` config
+
+270 questions were searched against the literature; **154 have a verified answer**
+(`is_verified_answer = true`). Each verified row has:
+
+| field | meaning |
+|---|---|
+| `answer` | the answer, in the item's own convention |
+| `answer_link` | the paper containing the answer |
+| `resolving_paper` / `resolving_theorem` | authors, title; exact theorem number |
+| `theorem_statement` | the theorem, quoted verbatim |
+| `derivation` | how that theorem answers the question *as literally phrased*; steps not stated in the paper are marked **INFERRED** |
+| `solution_sketch` | 1–3 paragraphs on how the paper proves it, from the paper's own outline |
+| `confidence`, `provenance`, `notes` | caveats |
+
+Unanswered rows keep their `status` (`STILL_OPEN`, `IMPROVED_BOUNDS`, `SOLVED_DIFFERENT`, …) and,
+where known, `current_bounds`.
+
+**Answer mix:** −1 (83), Yes (34), No (17), exact values (20) — including `7/32`, `(3-sqrt(5))/2`,
+`κ = 2`, finite minimal counterexamples such as `6`, `3` and `2`, and two `+infinity`.
+
+### Filter by provenance
+
+**124 answers are "clean"**: high confidence with none of these tags.
+
+| `provenance` tag | n | meaning |
+|---|---:|---|
+| `RECENT_PREPRINT_2026` | 19 | resolving paper not yet refereed |
+| `AI_ASSISTED_RESOLVER` | 9 | resolving paper declares AI/LLM involvement |
+| `COMPUTATIONAL` | 4 | rests on a computer(-assisted) check |
+| `NON_FINITE_ANSWER` | 2 | answer is `+infinity`; handle specially when grading |
+| `HAS_INFERRED_STEPS` | 79 | derivation contains an inferred step — usually the routine "proved for all *n*, hence −1" |
+
+### How it was built
+
+1. **Citation graph.** Semantic Scholar citations for all 12,019 pre-2020 source papers.
+2. **Candidate filter.** Citing papers whose title/abstract announces a proof, disproof or resolution.
+3. **Abstract screen.** `openai/gpt-6-sol` judged each (question, citing-paper) pair as `DIRECT`,
+   `SPECIAL_CASE`, `SIBLING`, `BOUNDS_ONLY` or `UNRELATED` — most candidates resolve a *different*
+   conjecture from the same source paper.
+4. **Full-text verification.** Claude Code subagents read each `DIRECT` candidate in full, checked the
+   main theorem against the exact question, traced priority to the first proof, and wrote the entry.
+5. **Citation check.** Every resolver arXiv ID was matched against arXiv author metadata.
+
+Verification changed many screen verdicts: disproofs mistaken for proofs, special cases, retracted or
+corrected proofs replaced by valid later ones, and "proofs" from predatory venues rejected.
+
+### Mis-transcriptions in the source data
+
+**19 of 270 searched questions (7%)** are flagged `MISTRANSCRIPTION`: the source paper misstated its
+conjecture — a dropped hypothesis, a typo, a reversed sign, a wrong constant, or a degenerate edge
+case (e.g. `n = 1`) — so the literal question is trivially answerable while the real problem is
+different. These are **excluded** from the verified answers. They **cluster by source paper**; a
+paper with one confirmed mis-transcription should be treated as suspect.
+
+13 further questions are flagged `AMBIGUOUS_QUESTION` (vacuous, about a proof *method*, or mixing an
+empirical and an asymptotic claim) and are also excluded.
+
+## Fields (numerical / yesno)
+
+`uid`, `arxiv_id`, `paper_url`, `primary_category`, `signal_type`, `question_category`, `format`,
+`route`, `question`, `answer_convention`, `distinguished_value`, `source_question`, `engine`, and for
+`numerical` also `math_intent`, `math_intent_secondary`.
 
 ## Caveats
 
-- **No answer keys.** Restating the warning above because it determines what this dataset is for.
+- **No answer key for most items** — see the warning above.
 - **`yesno` items are unverified claims**, not labelled true/false.
 - **Route B items are harder than their sources.**
-- **A small number of `arxiv_id` values are wrong**, inherited from the source dataset: for part
-  of that corpus the id came from a model-written field rather than the filename, so a few
-  questions are attached to the wrong paper.
+- **A small number of `arxiv_id` values are wrong**, inherited from the source dataset.
 - **`signal_type` is an extraction label**, not a verified claim that a problem is still open.
 
 ## Licensing
 
-Questions and derived fields: CC-BY-4.0. `source_question` is inherited from
-`amphora/ArXivOpenProblems`. Unlike that dataset, no verbatim paper excerpts are included here.
+Questions, labels and answer-key fields: CC-BY-4.0. `source_question` is inherited from
+`amphora/ArXivOpenProblems`; `theorem_statement` quotes short excerpts from the resolving papers,
+with attribution via `resolving_paper` and `answer_link`.
 
 ## Citation
 

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