Ripple: An Open, AI-Formalized Lean 4 Framework for Computing with CRNs
We present Ripple, an open, AI-formalized Lean 4 framework for the mathematics of computing with chemical reaction networks (CRNs) -- one extensible, machine-checked development that gathers several strands of the field into a single setting, and is built to grow. It formalizes: the theory of which real numbers a CRN can compute -- a single Lean definition of real-time CRN computation, the class of reals it captures, and the compilation pipeline (a GPAC / polynomial-ODE layer, a dual-rail compiler, and four stages down to large-population protocols) that realizes them, built so that adding a new number is a plug-in; three landmark population-protocol majority algorithms -- approximate, exact, and self-stabilizing exact majority; the stochastic-to-deterministic bridge, through three machine-checked versions of Kurtz's mean-field theorem; and two classical Turing-completeness results -- Bournez--Graca--Pouly for the deterministic GPAC and Soloveichik--Cook--Winfree--Bruck for stochastic CRNs. Each of these is verified to depend on exactly the three Mathlib foundational axioms, with no sorry. Along the way the formalization repaired genuine, previously unnoticed gaps in published proofs -- a compilation step that can transiently leave the unit interval, and an algebraic-number construction that silently rests on Catalan's conjecture -- and surfaced a sharp open problem about when a holonomic series admits an exact, rational-data polynomial-ODE encoding. The whole development is open and every proof is kernel-checked, so the results can be independently re-verified; and because it was written predominantly by AI agents using only publicly available models, the workflow that produced it can be reproduced with the same public toolchain.