Quantum Prime Spectral Theory (QPST)

Quantum Prime Spectral Theory (QPST) introduces a canonical operator-theoretic framework addressing the Hilbert–Pólya paradigm from a structural and rigidity-based perspective. Rather than postulating a spectral realization of the Riemann zeros a priori, the theory constructs a self-adjoint Hamiltonian whose arithmetic and Archimedean components arise as independent, intrinsically defined operator blocks. The framework begins with a canonical decomposition of the Hilbert space into arithmetic and Archimedean sectors, encoding respectively the prime logarithmic frequencies and the universal gamma-factor normalization. Nontrivial spectral behavior emerges only through admissible unitary delocalization and bounded Schur–Hadamard masking, which induce an intrinsic coupling between these sectors while preserving self-adjointness and canonical normalization. Using Gaussian test functions, the paper establishes a Weyl law for the zero-type spectral functional with universal leading coefficient 1 over 2 pi, shown to be entirely inherited from the Archimedean sector and rigid under all admissible deformations. After removal of the rigid Weyl contributions, the remaining spectral information admits an exact canonical decomposition, in which all nontrivial fine structure is carried by a single mixed arithmetic–Archimedean trace invariant. This mixed spectral channel is shown to constitute a complete rigidity invariant for the zero-type spectral functional. Under minimal hypotheses, equality of the mixed channel implies equality of the associated spectral data. Geometric and normalization arguments further select a unique admissible internal delocalization, corresponding to a rotation by pi over two, compatible with the geometry of the critical line. Finally, the canonical mixed spectral channel arising within QPST is identified, at the level of tempered distributions, with the mixed term appearing in the classical Riemann–Weil explicit formula. As a structural consequence of this identification and the rigidity principle, the zero-type spectral functional associated with the canonical QPST Hamiltonian agrees with its classical counterpart. No probabilistic assumptions, random matrix models, or semiclassical chaos hypotheses are invoked. All spectral features analyzed in this work arise from the intrinsic arithmetic–Archimedean architecture of the operator. Keywords Hilbert–Pólya conjecture; Riemann zeta function; spectral theory; explicit formula; Weyl law; operator theory; arithmetic–Archimedean interference; quantum chaos.