Quantum Prime Spectral Theory (QPST)

This work is part of a multi-volume series developing explicit arithmetic–archimedean Hamiltonians inspired by the Hilbert–Pólya philosophy.Across the six parts, we build, analyze and numerically explore operators whose internal spectral structure combines: Arithmetic blocks with eigenvalues of the form k log p, reflecting the Euler product; Archimedean components derived from the Mellin–type measure associated with the Γ-factor; Hybrid and nonseparable couplings that mix discrete prime-indexed modes, continuous scales, and dynamical blocks; Unitary and non-unitary evolutions probing different spectral regimes. The full series establishes universal statistical phenomena such as GOE-type local laws, critical spacing regimes, universal power-law tails (with exponents 3/2 and 3), Montgomery-type pair-correlation, and mixed “chaotic–arithmetic” spectral signatures. Each part develops a different layer of the global operator, offering a systematic progression from discrete phase–arithmetic models to continuous Hilbert–Pólya-type Hamiltonians. Part I–VI together form a unified program, introducing explicit constructions, rigorous operator-theoretic foundations, numerical experiments, and phenomenological interpretations toward an arithmetic spectral framework compatible with Hilbert–Pólya-type ideas.