Stroboscopic Resonance and Self-Referential Divisor Automata in Discrete Phase Lattices: A Hierarchical Farey-Sieve for Geometric Arithmetic

◉ ● DOI Title: **"Stroboscopic Resonance and Self-Referential Divisor Automata in Discrete Phase Lattices: A Hierarchical Farey-Sieve for Geometric Arithmetic"** ● Description: This work establishes a novel computational framework connecting discrete complex exponentials reⁱ²πko/r to stroboscopic resonance phenomena through a self-referential two-level divisor automaton. The research demonstrates that slow "ghost" trajectories in phase-sampled systems correspond to hierarchical resonances (+r/−r) generated by integer ratios between sequence elements, with positive resonances (+r) arising from direct Farey fractions and negative resonances (−r) from second-order ratio composition. Key contributions include: (1) a predictive algorithm for detecting phase-locked slow manifolds without exhaustive search; (2) identification of "desert" regions as gaps in low-denominator Farey sequences; (3) a bridge between Diophantine approximation, symbolic dynamics, and discrete geometric arithmetic via modular strobing conditions. The framework unifies number-theoretic divisor structures with dynamical systems theory (KAM tori, Poincaré sections) and provides computable tools for spectral factorization, signal processing, and resonance detection in discrete sampling systems.