Quantum Prime Spectral Theory (QPST)

This work presents a canonical operator theoretic framework inspired by the Hilbert Polya philosophy and centered on the Riemann explicit formula. A global self adjoint operator is constructed as the sum of two independent components an arithmetic operator with discrete logarithmic spectrum and an Archimedean operator with continuous spectrum generated by dilations on the positive real line.

The arithmetic component encodes the multiplicative structure of the integers and produces the von Mangoldt weights through the logarithmic derivative of its regularized spectral determinant. The Archimedean component accounts for the real place and yields the Gamma factor via finite part regularization of its continuous spectrum. Within this framework the explicit formula emerges canonically as an identity of regularized traces of the resolvent of the global operator in the classical Titchmarsh Mellin form.

The non trivial zeros of the Riemann zeta function do not appear as genuine eigenvalues but as spectral singularities of the resolvent reflecting the interaction between discrete arithmetic spectrum and continuous Archimedean spectrum. The self adjointness of the global operator enforces the spectral symmetry corresponding to the functional equation and singles out the critical line as the natural axis of symmetry.

No new claim concerning the Riemann Hypothesis is made. The contribution of this work is a conceptual clarification of the structural role of continuous spectrum spectral regularization and self adjointness in the operator theoretic interpretation of the explicit formula.