The Matthews Resonance: Deterministic Universal Capture Architecture with Multiple Physical Realizations

This record contains the final pre-print manuscript and reproducible numerical implementation of the Matthews Resonance framework. The formulation defines a deterministic nonlinear phase-locking mechanism for rational p:q ratios using a resonance-targeted sine coupling and a reduced Adler-type phase equation with scaling proportional to (p^2 + q^2). The framework derives explicit capture conditions, harmonic ladder closure, and falsification criteria. Numerical integration is performed in dimensionless form with long-horizon runtime and perturbation recovery tests across multiple coprime (p, q) configurations. The geometric realization termed the Matthew Curve is included as an interpretive embedding of phase-locked dynamics.

Certain implementation aspects are subject to a pending patent application.