Formalization of the Golay-Hopf Machine: A Unified Algebraic Framework for Hida, Iwasawa, and Yang-Baxter Structures

We present a unified algebraic framework, the “Golay-Hopf Machine,” which synthesizes four distinct mathematical structures: Golay coding theory, Hida theory, Iwasawa theory, and Yang-Baxter integrability. By defining a Hopf algebra structure on the binary Golay weights W = {0, 8, 12, 16, 24}, we show that: (1) Hida transitions correspond to the coproduct ∆, (2) Galois height corresponds to the counit ε, and (3) the weight complement w 7 → 24 − w acts as the antipode S satisfying S2 = id. We formally verify in Lean 4 that this structure satisfies the Yang-Baxter compatibility condition for heights and the Iwasawa logarithmic identity. All core algebraic results are verified with zero axioms and zero sorry statements. Finally, we sketch a roadmap for extending this framework to Anabelian geometry.