Datasets:

ulamai
/

SOTA-Math

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:

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.

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

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

AIM-AG research tasks

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

Counterexample-oriented variants

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

Loading

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

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

@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}
}