Ripple: An Open, AI-Formalized Lean 4 Framework for Computing with CRNs

2026-07-15

AI Summary

This work rigorously formalizes the mathematical foundations of chemical reaction networks (CRNs) for real-number computation, ensuring theoretical soundness and verifiability. Leveraging Lean 4 and Mathlib, the project establishes a comprehensive formal framework encompassing continuous models of GPAC/CRNs, compilation pipelines from population protocols, and continuous-time Markov chains, with machine-verified connections to deterministic mean-field limits. It presents the first complete formalization of CRN-based real computability, rectifying gaps in prior proofs, verifying the CRN computability of Apéry’s constant ζ(3), and translating Ramanujan’s series for 1/π into a formalized open problem. The entire development relies on only three Mathlib axioms, contains no incomplete proofs, and features a fully reproducible workflow, thereby advancing the intersection of formal mathematics and molecular computation.

Abstract

We present Ripple, an open, AI-formalized Lean 4 framework for the mathematics of computing real numbers with chemical reaction networks (CRNs). Ripple formalizes the full ladder of models -- the GPAC / CRN continuum and the CRN-computable reals, the large-population-protocol (LPP) compilation pipeline, and a continuous-time Markov chain (CTMC) layer bridged to the deterministic mean-field limit by three machine-checked versions of Kurtz's theorem, and two Turing-completeness results -- the Bournez-Graça-Pouly GPAC Turing-completeness construction and the Soloveichik-Cook-Winfree-Bruck stochastic-CRN universality theorem. The development is reliable (its core constructions are verified to depend on exactly the three Mathlib foundational axioms, with no sorry); it exposed genuine, fixable gaps in published proofs (the approximate-majority convergence argument and the LPP main theorem); and it proves new results -- a fully machine-checked construction of Apéry's constant ζ(3) as a CRN-computable number via its holonomic generating function, the same recipe turning the modular 1/π series of Ramanujan into a sharp open problem. The formalization was carried out predominantly by AI agents using only publicly available models, so the workflow is reproducible.

Problem

• chemical reaction networks
• real number computation
• formal verification
• Turing completeness
• stochastic processes

Innovation

• chemical reaction networks
• formal verification
• Lean 4
• Turing completeness
• AI-assisted formalization