The Hecke--Lerch Correspondence IV: The Global Operator and the Riemann Hypothesis

Part IV of "The Hecke-Lerch Correspondence" SeriesContext:This is the final paper in the "Hecke-Lerch Correspondence" series. It synthesizes the local constructions from Parts I (Rational), II (Quadratic), and III (Archimedean) into a unified global theory, culminating in the construction of a Hilbert-Pólya operator for the Riemann Zeta function.Abstract:This paper completes the local-global theory of the Hecke-Lerch correspondence, providing an explicit analytic construction of completed L-functions and a spectral reformulation of the Riemann Hypothesis.Building on the rational case in Part I, where reciprocal generating functions from the Lerch transcendent encode prime splitting in cyclotomic fields, and the imaginary quadratic extension in Part II, we incorporate the archimedean completion from Part III. Assembling these local kernels multiplicatively over all places, we construct a bounded normal operator $T_{\mathbb{A}}$ on the adelic Hilbert space $L^2(\mathbb{A}^\times/\mathbb{Q}^\times)$, whose Mellin action multiplies by $\Lambda(1-s)$.We prove strong operator convergence, factorization, and a distributional vanishing lemma tying zeta zeros to kernel orthogonalities. A geometric curvature density $K(t)$ interprets the spectrum, with archimedean hyperbolic negativity balanced by arithmetic spikes at zeros. The Riemann Hypothesis equates to spectral localization on the unitarity locus or curvature confinement to the critical line, furnishing the first explicit Hilbert-Pólya operator for zeta, extensible to general L-functions.Series Links:Part I: The Rational Case and Explicit Class Field TheoryPart II: Imaginary Quadratic Fields and the Analytic Construction of Class Field TheoryPart III: The Archimedean Completion and Gamma Factors