Spectral Basis Theorem

This entry presents and proves the Spectral Basis Theorem, the foundational result of Spectral Number Theory (SNT). The theorem reinterprets the classical Fundamental Theorem of Arithmetic as a spectral decomposition over a prime-logarithmic basis. In this formulation, every positive integer is viewed as a finite-support vector in an infinite-dimensional space, where each axis corresponds to a prime frequency and the components encode its harmonic activation. The result lays the groundwork for the broader SNT framework, which treats integers as waveforms, primes as basis modes, and arithmetic as resonance. This document is intended as both a mathematical reference and an accessible companion to the daily Xpost lectures.