A Time-Layered Variational Framework Unifying Optics, Mechanics, and Information

We revise and extend a time-layered variational framework that unifies opti- mization principles in optics, mechanics, and information. The framework sepa- rates internal time (local differential optimality and certificates: Snell’s law, Euler– Lagrange equations, constant hazard) from external time (global constraints: dead- lines, causality, speed-of-light bounds). Mathematically, we formalize external time by a time-dependent weighting w(t) and/or deadline constraints directly in the ac- tion functional. We give balanced derivations for optics (Fermat integral and Snell), mechanics (cycloid brachistochrone and the time-domain certificate¨ v+ Ω2v = 0), and information (maximum entropy-rate selecting the exponential law). A nor- malization map aligns the characteristic rates κ across domains and yields a toy example of triple equivalence. We report FDTD simulations (1D and 2D temporal refraction) that exhibit robust sideband redshift under external gating; tables in- clude p-values and gate widths. Non-idealities (smooth index steps and noise) are modeled, and an electro-optic (Pockels) experiment is proposed.