Huahe Theory: A Unified Framework of Natural Number Structure, Core Recursion, and Discrete Integrability

Unified Abstract (Brief Description):Huahe Theory is a comprehensive mathematical framework that reveals the underlying structure and evolutionary laws of natural numbers. It is built upon three mutually reinforcing pillars: The Fundamental Core Formula I – a unique decomposition of every odd integer N=2i3rR−1=2j3pQ+1N=2i3rR−1=2j3pQ+1, where R,QR,Q are coprime to 6 and the exponents r,pr,p obey strict mutual exclusivity. This formula defines the “Core” – the intrinsic algebraic skeleton of integers – and directly governs discrete systems such as the Collatz iteration, as well as analytic objects like Dirichlet series, generating functions, Mellin transforms, and continuum limits leading to elliptic PDEs. The Core Recursion Formula (FCR) – a unified recurrence An=αn+βn(An+1+γn)tAn=αn+βn(An+1+γn)t with t∈Rt∈R, which subsumes linear recurrences, continued fractions, quadratic recurrences, and nested radicals. Its continuum limit yields Painlevé I, while its bilinear (Hirota) form and Lax pair prove its integrability. A complete classification of the t=−1t=−1 family reveals both trivial (explicit product) and non‑trivial (discrete Painlevé) integrable cases. Error Compression and Transcendental Generation – the square recursion R(x2)≤(R(x)−1)2+1R(x2)≤(R(x)−1)2+1 for a sieve error function, equivalent to the Riemann Hypothesis and sufficient for the Goldbach and Legendre conjectures; and the algebraic nested-radical representation of ππ derived from the cosine half‑angle formula, showing how discrete iterations approach transcendental constants. The theory’s core tenet is “using the Core to govern Transformation, and using Transformation to verify the Core” – a philosophical and mathematical principle that ties together number theory, discrete integrable systems, and continuous analysis in a single, self‑consistent picture.