mjbommar/openlemma-v0.1-preference

OpenLemma v0.1: verified preference pairs

74,862 chosen/rejected pairs for post-training. The chosen answers are complete, verified derivations; each rejected answer has a flaw that the checker constructed and proved (a wrong step, a missing justification, an invalid rule). They are not part of the pretraining documents. input is the prompt, answer the chosen response, and record holds both responses, the flaw and its proof.

Provenance and verification

Every row comes from OpenLemma (code: MIT; the documentation in docs/ covers every source).

• Generation: procedural generators and gpt-6-luna (OpenAI, flex, reasoning effort low). Some upstream retrieval material was written by gpt-5.4-mini.
• Auditing only: gpt-6-sol.
• Checks:
  • verified: an independent checker, solver (axeyum), CAS route or truth table recomputed the answer;
  • constructed: the label is fixed by how the item was built (planted values, code corruptions, the sense a sentence was written for);
  • agreed: a model-proposed label that independent blind model checks agreed with, in a family whose gpt-6-sol audit is >= 95% (default) or 80-95% (extended, with audit_accuracy on every row);
  • prose: model-written explanatory text about verified facts.

Anti-copy packing. k-shot documents (k in {0, 1, 3, 8}) never reward copying the previous answer:

• open answers are pairwise distinct;
• label answers repeat the last demonstration's label only at the base rate 1/n, with labels balanced over the demonstrations;
• copy traps and symbol labels appear on 10% of documents each.

Batches v0/v1 were originally packed so that "any label but the last one" was always right; this release repacks those families.

Plain text. Documents in default/extended contain no control tokens. ALEA-style <|start|>/<|end|> boundaries are added at tokenization. Reasoning glyphs (① ∴ ∵ ⊢ ● ⚠ …) are ordinary characters. The special_tokens config keeps the original ALEA control-token renderings for replication.

Token counts use the ALEA tokenizer alea-p1-prefix-16k.

Licence and attribution

CC-BY-4.0. Attribute OpenLemma (https://github.com/mjbommar/openlemma). Upstream sources:

• axeyum (MIT OR Apache-2.0): curriculum, facts, scenarios, CAS and solver checks;
• OpenGloss v2.4 (CC-BY-4.0): lexical seeds and grounding;
• WordNet 3.0: rows derived from WordNet senses carry the WordNet licence, included as LICENSE-WordNet.txt;
• alea-quality-model: the 93 foundational reasoning constructs.

Generated text was produced with OpenAI models and is subject to the provider's terms.

Known limitations

• agreed labels are model-checked, not proven; read audit_accuracy and the per-family audits in the repository docs.
• OpenGloss v2.4 has known defects (about 10% of common senses invalid, unreliable etymologies). Etymology tasks are excluded, and seed senses were validity-checked for the newer runs.
• Two deontic constructs are thin (Permission Operator, Prohibition Operator).
• This release was published before any model was trained on it. It makes no claim about downstream gains.

Citation

@misc{openlemma2026,
  title  = {OpenLemma: verified, bottom-up synthetic data for logic, reasoning, algorithms and language},
  author = {Bommarito, Michael J.},
  year   = {2026},
  note   = {Version 0.1},
  url    = {https://github.com/mjbommar/openlemma}
}