Two Computable Invariants for Fourier Approximation of Piecewise-Smooth Signals

This technical note presents two linked, falsifiable diagnostics for Fourier approximations of piecewise-smooth periodic signals with jump discontinuities: an energy concentration invariant and a radius budget invariant. The note provides theorem statements, asymptotic mechanisms, falsification criteria, and numerical verification for canonical square-wave, sawtooth, and triangle controls. It is the primary scholarly artifact for this release; software and reproducibility code are provided in the related GitHub repository.