Spectral Arithmetic Framework Unified Mathematical Theory

This paper presents the complete mathematical theory of the Spectral Arithmetic Framework with rigorous proofs across four independent domains: number theory, cascaded networking, quantum process tomography, and autonomous robotics. Twenty-one hidden structures are identified and proven, including the Arithmetic Hamiltonian (HS-21): the free boson Hamiltonian H = Σ_p log(p) · a†_p a_p whose partition function is the Riemann zeta function, whose V₄ symmetry yields a conserved parity charge via Noether’s theorem, and whose parity conservation is equivalent to the Riemann Hypothesis. All conservation laws are explicitly classified: the Noetherian conservation of V₄ parity (from the Hamiltonian) is distinguished from the algebraic budget identities (R(k)+L(k)=1, uncertainty principles) that arise from normalization constraints. The spectral ghost (HS-8) is identified as the Goldstone boson of spontaneously broken V₄ parity.