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Ripple is an AI-formalized Lean 4 framework that rigorously formalizes the mathematics of computing real numbers using chemical reaction networks (CRNs), bridging various models and proving significant theorems in the process. It includes machine-checked versions of Kurtz's theorem and Turing-completeness results, while also identifying and addressing gaps in existing proofs. Notably, Ripple achieves a fully machine-checked construction of Ap茅ry's constant as a CRN-computable number, showcasing the potential of AI in formal mathematics.
AI agents have successfully formalized complex mathematical constructs, revealing gaps in existing proofs and proving new results, all within a reproducible framework.
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-Gra\c{c}a-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 Ap\'ery's constant {\zeta}(3) as a CRN-computable number via its holonomic generating function, the same recipe turning the modular 1/{\pi} 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.
