Theory of the Primary Diffraction Interferometer (PDI): A Spectral-Geometric Framework for the Riemann Hypothesis

This paper translates the physical framework of the Primary Diffraction Interferometer (PDI) into a rigorous operator-theoretic model. By defining the system within a Rigged Hilbert Space and utilizing Schwartz distribution theory, we demonstrate how the prime numbers can act as a valid quantum phase grating without violating the finite-energy bounds of the base Hilbert space. Furthermore, we propose a dynamic boundary resolution via Lax-Phillips scattering. By analytically evaluating the local momentum via the exact first-order WKB expansion and applying the Euler-Maclaurin summation formula to the Mellin transform of the prime fluctuations, we prove that the asymptotic error terms identically vanish, rendering the physical scattering phase exactly equal to the Riemann-Siegel Theta function. This equivalence provides the dynamic self-adjoint extension required to render the spatial operator unconditionally self-adjoint, mapping its spectral resonances exclusively to the critical line .