Match the verifier to the mathematical objective.
Mathematical post-training does not need one universal data format. Cheap exact answers, executable certificates, and open research exploration each optimize a different part of the capability stack.
AIME-style exact answer
Integer targets from 000–999, inexpensive deterministic reward, and difficulty from high school through graduate and research level.
Graduate + research RL
Proper verifier-backed environments with exact outputs or certificates, Harbor-compatible packaging, and blind trajectories.
Open research problems
No automatic verifier. Generate many rollouts, compare approaches, critique claims, and distill promising research reasoning.
Original task creation
Internally authored, unpublished problem families with controlled provenance and private evaluation boundaries.
1. AIME-style RLVR: the simplest strong verifier.
Each problem has one canonical integer answer from 0 to 999 and a zero-padded three-digit representation. The reward can be a deterministic exact match, making the format inexpensive to run at scale and free from judge-model variance.
def reward(model_output, golden_answer):
predicted = normalize_integer(model_output)
return 1.0 if predicted == golden_answer else 0.0The difficulty ladder is deliberately wider than the competition label suggests: standard AIME-style, AIME Hard, AIME-Graduate, and AIME-Researcher. The public AIME++ sample exposes 157 original problems across those four tiers, including 20 graduate and 5 researcher examples. The production catalog is much larger and can be mixed by domain, technique, answer distribution, or model difficulty.
Best uses
- High-throughput RLVR and rejection sampling with a stable binary reward.
- Difficulty curricula that retain one simple output contract from school to research-level mathematics.
- Evaluation slices resistant to verbosity, style, and judge-model preferences.
- Trajectory generation where reasoning can vary freely but the terminal reward remains exact.
2. Graduate and research Math RL environments.
These are proper RL environments rather than answer-only rows. A task can require an exact polynomial, a graph certificate, a structured JSON object, a rational-point witness, a proof artifact, or a sequence of stateful symbolic actions. The verifier recomputes invariants or checks constructive certificates.
The public Math-RL-Tasks sample pairs each suite's runnable environment with preserved blind-run material, prompts, submissions, verifier receipts, concise reasoning summaries, limitations, and trajectories. It spans areas including algebraic geometry, arithmetic geometry, combinatorics, topology, probability, spectral analysis, and research-style exact computation.
- Prompt: original mathematical specification and exact submission schema.
- Environment: dependencies, state, tools, resource policy, and deterministic seed.
- Action: answer, certificate, code, graph, or structured artifact.
- Verifier: recompute the invariant, validate the witness, enforce schema, and return criterion-level reward.
- Evidence: freeze the submission, retain the receipt, and package a concise trajectory without private hidden chain-of-thought.
In selected internally tested production environments, GPT-5.6 Sol Pro blind-pass rates typically fall in a useful 60–90% range. That is an observed working range and environment-design target—not one aggregate score for the heterogeneous public sample. Every shipped result should retain its own protocol, task count, verifier version, and limitations.
3. Research-level mathematics without an automatic verifier.
Some of the most valuable problems are open, research-level, or too expensive and judgment-heavy for a deterministic grader. Their value comes from generating many independent approaches and turning the resulting search distribution into post-training data.
The SOTA-Math sample shows the format. A production workflow can run a large rollout pool, cluster methods, identify recurring lemmas and bottlenecks, compare candidates with expert review and cross-rollout consistency, and retain both promising partial progress and failed approaches.
| Stage | Output |
|---|---|
| Parallel exploration | Diverse proof attempts, computational experiments, counterexample searches, and literature-free approaches |
| Cross-rollout analysis | Claim graph, shared lemmas, contradiction map, novelty clusters, and uncertainty labels |
| Expert review | Correctness assessment, promise rating, missing obligations, and research direction |
| Post-training views | Critiques, revisions, preference pairs, method comparisons, partial-proof continuation, and distilled solutions where available |
Evaluation uses expert review and uncertainty-aware adjudication instead of an automatic binary reward. The release should make reviewer identity or qualification, adjudication process, uncertainty, and unresolved obligations explicit.
Original unpublished problems are the core asset.
Ulam AI creates the core mathematical problems internally. They are original and unpublished before controlled release, which supports genuine novelty, private holdouts, and model-specific hillclimbing rather than benchmark memorization.
A separate product line can convert published problems and solutions into RLVR or trajectory formats where provenance and rights permit. This includes curated collections such as original Erdős problems. Published-source material remains clearly labeled and isolated from the internally authored unpublished corpus.
- Stable source lineage and authorship records for every item.
- Similarity and contamination checks across public corpora and client training data.
- Separate train, development, and private evaluation boundaries.
- Versioned solutions, verifiers, expert reviews, and exposure logs.
What a Ulam Math program can deliver.
- Custom AIME-style exact-answer collections by tier, domain, technique, answer distribution, and frontier difficulty.
- Graduate and research Harbor-compatible RL environments with exact verifiers, certificates, tests, seeds, and blind-run evidence.
- Open research problem sets with multi-rollout protocols, expert review, claim graphs, critiques, and preference data.
- Safe trajectory packages: prompts, actions, tool results, concise rationales, rewards or reviews, failure labels, and repairs.
- Private evaluation releases and renewable hillclimb batches selected from the client's own model failure surface.
Public samples: AIME++, Math RL Tasks, and SOTA-Math. Machine-readable evidence policy: JSON.