A Formal Dynamical Framework for the ϕ–Zeno–CORDIC Prime Resonance System

We present a formal mathematical and hardware-level framework unifying continuous torus ergodic dynamics, Binet’s Fibonacci wave conjugate decay, and discrete fixed-point CORDIC execution. By defining an irrational rotation vector on the 2-torus T2derived from the Golden Ratio ϕ and ln ϕ, we construct a Zeno phase map Θ that models logarithmic spiral contractions. We project this continuous system onto a discrete K-digit fixed-point space Q2 K governed by a CORDIC shift-and-add operator CK. We prove that prime indices p induce an explicit Diophantine quadrature lock into (2m + 1)π 2, triggering an ultrametric zero-node collapse in the machine basin Z2 K. Conversely, we establish a composite phase dispersion lemma showing that multiplicative indices destroy quadrature alignment through phase factor convolution.This establishes a structural, hardware-executable dynamical invariant characterizing prime indices.