Lemma Labs

We build reinforcement learning environments that teach models mathematics. In these environments a model attempts a proof, a formal verifier checks the reasoning, and the result becomes a reward signal that guides the next attempt. Choosing the problems, supplying the right definitions and tools, and making sure the reward reflects the mathematics we actually want learned is the work.

Mathematics is unusually valuable for this because it combines difficult reasoning with precise feedback. A verifier can tell you whether a proof establishes a formal statement. It cannot tell you whether the statement was worth proving. That judgment is the hard part, and it is where we concentrate.

Every other company supplying environments spreads across law, finance, customer support, coding, and mathematics. Breadth pushes them toward shared production processes and shared evaluation. Advanced mathematics needs specialist judgment at nearly every stage: reading a theorem, finding its assumptions, tracing its dependencies, and deciding whether a task will teach anything at all. We do mathematics and nothing else.

Our work

We recompiled a random sample of one of the larger public Lean proof corpora against current Lean and Mathlib. A third of it is defective on arrival, in a way that had gone unreported.

We also expanded five research proofs into complete dependency graphs. Apéry's proof that ζ(3) is irrational rests on 885 theorems beyond the base library, reaching thirty-six levels deep.

Essays

Team

IMO gold and silver medalists from MIT and Princeton, mathematics PhD students at Stanford, and machine learning researchers who turn research into working systems. We have relationships with students and faculty at MIT and Princeton that connect us to the communities where this mathematics is developed.

Those are reasons to take us seriously, not evidence that our environments work. We intend to measure whether our mathematics is correct, how much of the subject we cover, whether trained models improve on problems they have never seen, and what each accepted example costs to produce. We'll report the results either way.

Contact

[email protected]