The Matthew Curve: A Unified Geometric Framework for Black-Hole QPOs and Phase-Locked X-Ray Polarization

Description This paper presents the mathematical foundations of the Matthews Resonance Framework, a geometric theory of rational phase locking derived from two periodic phases, a coprime resonance ratio, and a minimal first-harmonic coupling potential. The framework shows that the state space of two periodic phases is naturally the two-dimensional torus (T²), from which the phase residual, its geometric gradient, and the effective coupling law arise. A central result is the decomposition of the effective coupling, K_eff = ε(m² + n²), where ε represents the physical coupling strength and the factor (m² + n²) is determined entirely by the selected coprime resonance ratio. From this construction, the Matthews Residual ODE, phase-capture condition, potential well, and Bézout harmonic ladder are derived as mathematical consequences rather than postulated assumptions. The paper further extends the geometric framework to continued-fraction hierarchies, showing that the Fibonacci sequence is one member of a broader family generated by quadratic irrationals. Possible applications to black hole quasi-periodic oscillations (HFQPOs), neural oscillations, rotating machinery, electronic oscillators, and other coupled periodic systems are discussed as testable physical hypotheses. This work is intended as a mathematical foundation for the Matthews Resonance Framework and provides the theoretical basis for subsequent observational and experimental investigations.