Homometric Phonon Filters: Identical Periodic Diffraction and Distinct Energy Transmission

This mathematical physics paper studies what diffraction and bulk phonon dispersion can determine about transmitted energy. Exact constructions have identical periodic pair diffraction, and even identical complete Floquet relations, but different contacted transmission and thermal conductance. The principal results include quantitative inference obstructions, weak-contrast filtering bounds, and complete acoustic-conductance comparisons. In three dimensions, rigorous angular integration reveals a logarithmic low-temperature contribution governed by grazing waves. A diamond-based design admits a conditional fabrication-tolerance certificate. Passive-loss theorems establish finite coherent length optimization, while absorption budgets bound thermal re-emission from floating reservoirs. The proofs use transfer-matrix identities, scattering moments, uniform asymptotics, rational certificates, and variational reservoir reduction. Experimental realization, calibrated full vector-elastic diamond transport, and joint material-and-loss optimization remain open.