Hua‑He Core Theory: Two Deep Paths from the Core Formula 2 i 3 r G - 2 j 3 t Q=2 Differential Equations and Analytic Number Theory

The core formula 2i3rG−2j3tQ=2 is the unique axiom of Hua‑He Core Theory. This paper explores two independent deep directions: (1) a continuous/differential approach, which reveals that the exponents must evolve synchronously and leads to an integrable flat geometry; (2) an analytic‑number‑theoretic approach, where the formula is interpreted as a Diophantine equation and studied via lower bounds for linear forms in logarithms (Baker’s theorem). The two paths together show that the same elementary identity simultaneously governs discrete dynamics (Collatz type), continuous conserved quantities, and the scarcity of solutions to exponential Diophantine equations (Fermat, Catalan, Beal). A unified summary table illustrates the multiple levels at which the core formula operates.