# Lean

Physics-informed neural networks could previously show only small residuals at sampled points, not how large the error is over the continuous domain. This pipeline has a network search the Babenko equation in conformal coordinates for high-amplitude gravity–capillary waves, and staged neural corrections, Fourier projection and Newton polishing cut the maximum sampled residual by about seven orders of magnitude. The candidate is then frozen as an explicit trigonometric polynomial, interval arithmetic yields a rigorous uniform bound on the continuous residual, and Lean machine-checks the deductions from that numerical bound to the final residual theorem.

Physics-informed neural networks could previously show only small residuals at sampled points, not how large the error is over the continuous domain. This pipeline has a network search the Babenko equation in conformal coordinates for high-amplitude gravity–capillary waves, and staged neural corrections, Fourier projection and Newton polishing cut the maximum sampled residual by about seven orders of magnitude. The candidate is then frozen as an explicit trigonometric polynomial, interval arithmetic yields a rigorous uniform bound on the continuous residual, and Lean machine-checks the deductions from that numerical bound to the final residual theorem.

Axiom released a Lean formalization of the Stacks Project, turning the roughly 7,500-page open reference that lays the foundations of modern algebraic geometry into a single machine-checked library of about 16 million lines of code. Algebraic geometry had lacked matching Lean infrastructure: a contemporary proof may invoke a long chain of theorems across commutative algebra, schemes, sheaves, cohomology and algebraic stacks before its core argument begins. The library keeps the original permanent tags so researchers and AI agents can import definitions and theorems directly, and the formalization surfaced dozens of statements needing local refinement.

Axiom released a Lean formalization of the Stacks Project, turning the roughly 7,500-page open reference that lays the foundations of modern algebraic geometry into a single machine-checked library of about 16 million lines of code. Algebraic geometry had lacked matching Lean infrastructure: a contemporary proof may invoke a long chain of theorems across commutative algebra, schemes, sheaves, cohomology and algebraic stacks before its core argument begins. The library keeps the original permanent tags so researchers and AI agents can import definitions and theorems directly, and the formalization surfaced dozens of statements needing local refinement.

Proving software correct is often more expensive than writing it, and finished verification work rarely becomes a reusable theorem library. VeriBase treats code and its associated theorems as one declaration family and builds Lean libraries automatically through proving, generalization and harvesting, updating dependent declarations as interfaces change. A general library built from 4,191 independent programs raised average task success from 63.0% to 73.0% across three single-file benchmarks and three backend models, while cutting average completion time by 13.2%.

Proving software correct is often more expensive than writing it, and finished verification work rarely becomes a reusable theorem library. VeriBase treats code and its associated theorems as one declaration family and builds Lean libraries automatically through proving, generalization and harvesting, updating dependent declarations as interfaces change. A general library built from 4,191 independent programs raised average task success from 63.0% to 73.0% across three single-file benchmarks and three backend models, while cutting average completion time by 13.2%.

In Lean 4 theorem proving that depends on project context, failed proofs used to be discarded as a final filter result, and independent sampling restarted later attempts nearly from scratch. This search framework draws one candidate each from a general-purpose reasoning model and a Lean-specialized prover, uses compiler errors in a pairwise comparison to pick the current-best proof, accepts a revision only when the comparison judges it better, and resamples when refinement stagnates. On seven real Lean 4 projects from miniCTX-v2, within the pass@32 budget it improves average pass rate by 12.8 percentage points over the two base models' average pass@32 while cutting LLM calls by 21.9%.

In Lean 4 theorem proving that depends on project context, failed proofs used to be discarded as a final filter result, and independent sampling restarted later attempts nearly from scratch. This search framework draws one candidate each from a general-purpose reasoning model and a Lean-specialized prover, uses compiler errors in a pairwise comparison to pick the current-best proof, accepts a revision only when the comparison judges it better, and resamples when refinement stagnates. On seven real Lean 4 projects from miniCTX-v2, within the pass@32 budget it improves average pass rate by 12.8 percentage points over the two base models' average pass@32 while cutting LLM calls by 21.9%.

The work reframes method selection in formal proving as an applicability problem and introduces self-advertisement: in one batched call, each of 82 Method Contracts proposes a target, an action, and required conditions for the current goal, and vague or unsupported proposals are demoted before ranking; it reaches 95.0% hit@5 on Putnam 2015–2025 versus 84.2% for the strongest reranker, and 91.7% versus 88.3% on IMO ProofBench.

The work reframes method selection in formal proving as an applicability problem and introduces self-advertisement: in one batched call, each of 82 Method Contracts proposes a target, an action, and required conditions for the current goal, and vague or unsupported proposals are demoted before ranking; it reaches 95.0% hit@5 on Putnam 2015–2025 versus 84.2% for the strongest reranker, and 91.7% versus 88.3% on IMO ProofBench.