Unified Substrate Theory’s Harmonic Law Monograph develops the foundational scaling relation as the structural signature of systems governed by a discrete log-periodic substrate, and demonstrates how this dual polynomial–geometric form emerges naturally from the substrate field equation . Drawing on boundary coherence, regeneration frequencies, and curvature response, the monograph shows how the combined additive envelope and multiplicative factor generate the universal harmonic architecture underlying physical excitations across scales. It then compiles cross-domain empirical evidence—from planetary spacing and SPARC rotation curves to the universal acceleration scale, microphysical excitation ladders, seismic cascades, CMB lensing, SN Ia redshift harmonics, black hole mass distributions, and BAO β-silence—revealing that the same discrete frequency set recurs with high statistical significance in dynamically active sectors while vanishing in fossil regimes. The result is a unified mathematical–physical framework in which the harmonic law operates as a cross-scale organizing principle, and in which the repeated empirical recovery of the same β-spectrum provides a coherent evidential case for UST as a candidate universal substrate theory.