Resonant Operator Calculus (ROC): A Dimension-Independent Architecture for Stable, Directionally-Selective Discrete Dynamics

This work presents a spectral operator framework for discrete wave propagation on periodic lattices that achieves exact orthogonal decomposition into directional propagation modes. The method addresses the fundamental limitation of classical finite impulse response (FIR) filters, which inevitably exhibit spectral leakage due to the uncertainty principle in harmonic analysis.
The framework partitions the frequency torus into three disjoint sets (forward, neutral, backward propagating modes) and constructs projection operators that commute with spatial translation. This enables closed-form expressions for iterated dynamics and unconditional stability guarantees based solely on modulation coefficients.
Key contributions:
Rigorous proof of exact orthogonal channel decomposition
Closed-form iteration formulas with dimension-independent validity
Numerical validation demonstrating machine-precision accuracy (residual errors below 1e-14)
Application to one-way waveguides with backward suppression to 1e-32
The framework is applicable to metamaterial simulation, directional antenna arrays, optical isolators, and any discrete wave system requiring long-term numerical stability with directional selectivity.