Foundations of Spectral Mechanics—Supplemental Material (Full Series)

This document provides the foundational derivations, proofs, and mathematical structures supporting the full research program of Spectral Mechanics—a unified framework for a self-closed, background-free flat spacetime in a (E, p) manifold. It establishes that energy and momentum are the independent observables of physical systems, while space and time emerge as dependent accumulated functionals arising from their evolution. The supplemental material formalizes the spectral interval, Lorentzian structure, spectral phase coherence, Lorentz invariance, Born reciprocity, and other supporting elements formalising a background-free spacetime field within the spectral manifold. It serves as the complete technical reference for all current and future papers in the Foundations of Spectral Mechanics series. Version note: This is the full-series supplemental; future updates will extend it as the Spectral Mechanics framework evolves.