README.md

---
license: mit
language:
- en
pretty_name: "SOTA Math: Ulam.ai Research Problem Showcase"
size_categories:
- n<1K
task_categories:
- question-answering
tags:
- mathematics
- theorem-proving
- reinforcement-learning
- research
- tool-use
configs:
- config_name: showcase
  default: true
  data_files:
  - split: train
    path: data/showcase.jsonl
- config_name: erdos_variants
  data_files:
  - split: train
    path: data/erdos_variants.jsonl
- config_name: aim_ag_tasks
  data_files:
  - split: train
    path: data/aim_ag_tasks.jsonl
- config_name: counterexample_variants
  data_files:
  - split: train
    path: data/counterexample_variants.jsonl
- config_name: aim_ag_rl_tasks
  data_files:
  - split: train
    path: rl/data/public_tasks.jsonl
- config_name: aim_ag_rl_episodes
  data_files:
  - split: train
    path: rl/data/curriculum_train_hf.jsonl
  - split: validation
    path: rl/data/curriculum_validation_hf.jsonl
  - split: test
    path: rl/data/curriculum_test_hf.jsonl
- config_name: aim_ag_exact_benchmarks
  data_files:
  - split: train
    path: rl/data/exact_benchmark_train.jsonl
  - split: validation
    path: rl/data/exact_benchmark_validation.jsonl
  - split: test
    path: rl/data/exact_benchmark_test.jsonl
- config_name: aim_ag_frontier_eval
  data_files:
  - split: test
    path: rl/data/frontier_eval_public.jsonl
---

# SOTA Math: Ulam.ai Research Problem Showcase

This public release contains 20 internally authored, research-grade mathematics tasks selected to demonstrate the style of problem design and evaluation developed at [ulam.ai](https://ulam.ai):

- 5 new problems inspired by Erdős problems;
- all 10 problems in the AIM-AG research sample, with milestone-based RL task definitions; and
- 5 bounded counterexample or near-counterexample searches inspired by major conjectures.

These are candidate research problems, not a collection of known-answer exercises. A finite literature search cannot establish global novelty, and mathematical status can change. Proofs, disproofs, and novelty claims require current specialist review.

## Why this sample

The sample favors problems with crisp statements, nontrivial but checkable baseline arguments, multiple routes to progress, and explicit standards for computational evidence. It is intended to show prospective research and training partners what a larger private Ulam.ai collection can support: long-horizon reasoning, exact tool use, counterexample search, milestone credit, critic training, and expert-gated evaluation.

## Contents

| Configuration | Rows | Purpose |
|---|---:|---|
| `showcase` | 20 | Uniform, buyer-friendly schema across all three streams |
| `erdos_variants` | 5 | Full-fidelity selected Erdős-inspired records |
| `aim_ag_tasks` | 10 | Full research context, references, milestones, and failure conditions |
| `counterexample_variants` | 5 | Full-fidelity finite search and certification tasks |
| `aim_ag_rl_tasks` | 10 | Policy-visible RL task prompts |
| `aim_ag_rl_episodes` | 53 | Type-stable public curriculum: 18 train, 14 validation, 21 test |
| `aim_ag_exact_benchmarks` | 33 | Public exact-computation prompts: 11 per split |
| `aim_ag_frontier_eval` | 10 | Public full-task evaluation prompts |

The `rl/` directory also includes the full-fidelity public curriculum file, public fixtures, schemas, and reward/curriculum configuration. The Hugging Face configuration uses a lossless type-stable view because the original file represents exact prompts as strings and frontier prompts as objects. It records the original field inventory and JSON-encodes mixed or structured values, so every production-format public episode can be reconstructed. It intentionally excludes hidden grader records, hidden targets, hidden fixtures, calibration answers, and expert reviews.

The full AIM-AG research records and RL prompts are two public views of the same ten tasks. Their conjecture and definitions are synchronized. RL instructions, required deliverables, and allowed-tool wording intentionally remain production-oriented and may differ from the client research record. All included prompts and fixtures are public; train/dev/eval labels are organizational splits, not secrecy or holdout claims.

## Selected problem families

### Erdős-inspired variants

1. Bounded-defect rigidity for distinct subset sums
2. Profile-balanced sunflower threshold
3. Second-order certificate complexity of a squarefree-free interval
4. Median pinned-distance conjecture
5. RMS-superlevel concentration for Littlewood polynomials

### AIM-AG research tasks

1. A single simple branch among 321,489 planes
2. Lines as the largest Noether-Lefschetz loci on Fano threefolds
3. Generic maximal-rank edge maps for tree-glued varieties of minimal degree
4. A conductor bound for free anticanonical degrees on toric Fano varieties
5. Arithmeticity of vanishing cohomology in high-power linear systems
6. An integral Segal theorem for toric targets
7. A factorial bound for Rost nilpotence
8. A character-valued logarithmic gluing formula
9. A cyclotomic Hasse tower for K3 moduli
10. Logarithmic TP as a tame-ramification detector

### Counterexample-oriented variants

1. A closest joint Giuga-Carmichael local fit
2. A subdivision-poor five-critical graph at the open Hajós boundary
3. Interior hot spots on a convex lattice domain
4. The most dangerous bounded 14-runner instance
5. A bounded King-Tollu-Toumazet coefficient search

## Loading

```python
from datasets import load_dataset

showcase = load_dataset("ulamai/SOTA-Math", "showcase", split="train")
aim_ag = load_dataset("ulamai/SOTA-Math", "aim_ag_tasks", split="train")
episodes = load_dataset("ulamai/SOTA-Math", "aim_ag_rl_episodes")
```

The default `showcase` configuration uses one stable schema. The full-fidelity configurations preserve stream-specific metadata and should be loaded separately.

## Uniform showcase schema

Each of the 20 `showcase` rows has:

- stable identifiers and stream labels;
- title, domain, task type, and difficulty;
- a complete prompt and provenance-oriented inspiration field;
- rationale and expected output;
- research-status and verification caveats;
- RL readiness, milestones, and quality signals; and
- per-record MIT license metadata.

The machine-readable schema is in `schema/showcase.schema.json`.

## Evaluation policy

Exact finite subproblems may be checked with reproducible computation and machine-verifiable certificates. Frontier claims are expert-gated. Numerical evidence, finite-field experiments, dimension counts, or literature summaries are not silently upgraded to complete proofs. Strong partial progress is a valid outcome when its scope and remaining gaps are explicit.

The release contains no canonical solutions to the 20 headline problems. Public exact episodes are scaffolding for tool and evidence discipline, not answer keys to unresolved terminal conjectures.

## Intended uses

- evaluate research-agent decomposition and gap control;
- train or assess tool-augmented mathematical reasoning;
- prototype milestone-based process rewards;
- study exact-certificate generation; and
- assess fit for a larger private Ulam.ai research-problem program.

## Out-of-scope uses

- treating a model response as a verified mathematical result without review;
- claiming that every problem is globally novel or still open without a current literature refresh;
- using public episodes as a secret held-out benchmark; or
- inferring performance on all of mathematics from this deliberately small showcase.

## Creation and provenance

The problem statements and task packaging were authored internally at Ulam.ai. Inspiration labels identify the mathematical lineage of variants; they do not claim that the new task is the canonical statement of the cited conjecture. See `DATA_STATEMENT.md` and `MANIFEST.json` for selection, provenance, and integrity details.

## License

Copyright (c) 2026 ulam.ai. Released under the MIT License. See `LICENSE`.

Bibliographic citations and names of third-party mathematical results remain attribution facts about their respective sources; the MIT grant covers the Ulam.ai-authored dataset content and packaging.

## Citation

```bibtex
@dataset{ulamai_sota_math_2026,
  author    = {{ulam.ai}},
  title     = {SOTA Math: Ulam.ai Research Problem Showcase},
  year      = {2026},
  publisher = {Hugging Face},
  url       = {https://huggingface.co/datasets/ulamai/SOTA-Math},
  version   = {0.1.0}
}
```