Pure Time Theory – Chapter V - A Spectral–Geometric Route to the Riemann Hypothesis

Abstract This article presents the definitive resolution of the Riemann Hypothesis (RH) within the Pure Time Theory (PTT) framework, replacing and superseding my earlier preprint (*The Riemann Hypothesis as a Derived Theorem in Pure Time Theory*, v24). The v24 version formulated RH as a theorem inside an axiomatic system T (axioms T1–T5), showing that injectivity constraints on temporal phases force all non-trivial zeros of zeta(s) onto the critical line. That work was an essential first step, but it remained abstract and axiomatic. The present article (from Intuition to Genesis Chapter V v21) completes the loop opened by Chapters I–III of the Genesis of PTT. Building on the zero-induced geometry developed earlier, the proof now proceeds in a purely analytic and geometric way: 1. From an optical slice we construct a compact, odd boundary datum W. 2. Using Paley–Wiener synthesis, we build entire functions consistent with W. 3. A canonical separation isolates the Archimedean background, on-line zeros, and off-line factors. 4. Hardy–Smirnov rigidity enforces constancy of the comparison quotient. 5. Injectivity and the intrinsic flux–neutral gauge eliminate the off-line factor, proving that all zeros lie on Re(s)=1/2. This new derivation abandons the external axiomatic container and works directly with the geometry encoded by the zeta zeros themselves. It yields a fully self-contained, non-circular proof of RH that is compatible with classical analytic number theory while rooted in the modal architecture of PTT. The transition marks the maturation of PTT from a speculative axiomatic program into a concrete, structural demonstration that RH is a necessary consequence of zero-induced geometry.