README.md

---
license: cc-by-4.0
task_categories:
- question-answering
language:
- en
tags:
- mathematics
- arxiv
- open-problems
- research-level
size_categories:
- 100K<n<1M
configs:
- config_name: numerical
  data_files:
  - split: train
    path: data/numerical-*.parquet
- config_name: yesno
  data_files:
  - split: train
    path: data/yesno-*.parquet
- config_name: answers
  data_files:
  - split: train
    path: data/answers-*.parquet
---

# ResearchMath-2-6Sol-Rewrite

**534,298 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 | contents |
|---|---:|---|
| `numerical` | **140,248** | answer is a single number; `math_intent` labelled for 35,274 of them |
| `yesno` | **394,050** | answer is Yes or No |
| `answers` | **270** (154 verified) | literature search results, with link, theorem, derivation and solution sketch |

## v2 — full coverage (this version)

**v1** (tag [`v1`](https://huggingface.co/datasets/amphora/ResearchMath-2-6Sol-Rewrite/tree/v1), 97,497 items) was a
budget-limited subset: questions were rewritten in order of expected numerical yield — conjectures first,
then `math.CO` and `math.NT` — so analysis, probability and physics were heavily under-represented.

**v2 rewrites every question in `amphora/ArXivOpenProblems`** (534,298 of 534,309 rows) with the *same* model,
prompt and settings as v1 (`openai/gpt-6-sol`, batch tier, temperature 0.2). v1's items are kept unchanged;
`source_run` records where each row came from.

Category share of the `numerical` config relative to the source (1.00× = proportional):

| category | v1 | v2 |
|---|---:|---:|
| `math.CO` | 3.29× | **1.62×** |
| `math.NT` | 1.60× | 1.39× |
| `math.AP` | 0.47× | **1.01×** |
| `math.PR` | 0.34× | **0.80×** |
| `math.DS` | 0.27× | **0.83×** |
| `math.OC` | 0.34× | 0.49× |

The selection bias is gone. What remains is a **format** bias: route B ("smallest counterexample, −1 if none")
suits "for all *n*" statements, so combinatorics converts to a number 42.5% of the time and optimization 12.9%.
For a category-balanced numerical benchmark, sample from this set by `primary_category`.

On 1,872 questions rewritten in both runs, 89% got the same format and 93.6% of numerical pairs the same
`distinguished_value`; format flips were symmetric (101 vs 104), i.e. noise on borderline questions, not drift.

## ⚠️ Most items have no answer key

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

`route` records the construction used for numerical items:

| route | construction | share |
|---|---|---:|
| **B** | minimal counterexample: "smallest *p* at which the property fails; −1 if none" | **79.0%** |
| E | size of the exceptional set (0 = the original's "yes") | 8.8% |
| A | the question already asked for a quantity | 5.8% |
| C | sharp constant in an inequality | 2.1% |
| D | critical value of a continuous parameter | 2.2% |
| G | count of a finite classification | 1.8% |
| F | minimal/maximal multiplicity or order | 0.4% |

`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?".

## `math_intent` (numerical config)

What each question is *about* — orthogonal to `question_category` (its logical shape). Assigned by
`openai/gpt-6-sol` from the source question; labelled for the 35,274 numerical rows of v1 only (`null` elsewhere); the counts below are for those rows.

| math_intent | n | share |
|---|---:|---:|
| 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`, `source_run`, and for
`numerical` also `math_intent`, `math_intent_secondary`.

## Caveats

- **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.**
- **`numerical` still leans toward combinatorics** — a property of the conversion, not of selection (see v2 above).
- **636 fields had LaTeX swallowed as a JSON escape** (e.g. `\\theta` read as a tab) and were restored deterministically.
- **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, 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

```
@article{son2026researchmath,
  title={ResearchMath-14K: Scaling Research-Level Mathematics via Agents},
  author={Son, Guijin and Yi, Seungyeop and Gwak, Minju and Ko, Hyunwoo and Jang, Wongi and Yu, Youngjae},
  journal={arXiv preprint arXiv:2605.28003},
  year={2026}
}
```

## Collaborations

I'm interested in creating larger datasets to train open models for research-level math.
If you are interested let me know. ([guijin.son@snu.ac.kr](mailto:guijin.son@snu.ac.kr))