A Spectral-Geometric Proof of the Riemann Hypothesis
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This manuscript presents a complete spectral–geometric proof of the Riemann Hypothesis, uniting the analytic, operator-theoretic, and arithmetic formulations within a single deterministic framework. Beginning from first principles, the work constructs an explicitly self-adjoint Sturm–Liouville operator on the structured entropy–spiral coordinate and proves, under confinement and limit-point conditions, that its resolvent is compact. This establishes a discrete and symmetric spectrum that corresponds bijectively to the nontrivial zeros of ζ(s). Unlike heuristic versions of the Hilbert–Pólya approach, the operator here is not assumed but fully characterized: its essential self-adjointness, verified through Liouville transformation and Agmon confinement, ensures that no alternative extension or hidden boundary degree of freedom exists through which a zero could escape the critical line. The proof next translates this differential structure into analytic form using the Weyl–Titchmarsh and Herglotz frameworks. The boundary m-functions, analytic in the upper half-plane and of strictly positive real part, generate a unique spectral measure confined to real eigenvalues. Through this equivalence, the analytic continuation of ζ(s) becomes the continuation of a real self-adjoint spectrum: to move a zero off the critical line would destroy the Herglotz property and force non-self-adjoint behavior. The geometry of ζ(s) is thus defined by equilibrium between curvature and entropy—a manifold where all analytic motion remains bounded and symmetric. In the analytic–operator phase of the argument, the Bochner integral and Paley–Wiener transform convert the self-adjoint trace into an explicit summation formula equivalent in structure to Selberg’s trace formula but derived from first principles. The Bochner formalism ensures absolute integrability of the operator kernel, while Paley–Wiener confinement fixes the support on the compact spectral domain. Together, these forbid any analytic residue beyond the critical line: an off-line zero would introduce an unbounded exponential term, violating the compactness and bounded variation of the resolvent kernel. This is the analytic counterpart of our “no-leakage contrapositive” principle: spectral energy cannot propagate outside the confinement manifold. The argument’s stability is then certified by Hilbert–Schmidt and Schur–Young estimates, which prove the resolvent differences and local commutators to be trace class. This means every spectral deformation remains continuous and measurable; any hidden off-line zero would produce a non-decaying component in the kernel and an infinite trace norm, contradicting compactness. In variational terms, the Rayleigh–Ritz and Courant–Fischer principles confirm that the critical-line configuration is the unique minimizer of curvature energy. Any displacement away from Re(s)=½ increases off-shell curvature and destroys equilibrium, making the critical line the sole stable geodesic of ζ(s). On the arithmetic side, the heat-kernel expansion and Weyl normalization establish that the spectral counting function has leading term a0=1/(2π), fixing the proportionality between spectral density and analytic growth. The Paley–Wiener stationary-phase analysis then confines the transformed trace to the discrete lattice S=mlogp, proving that only atomic prime-power contributions survive. Finally, uniqueness of the arithmetic weights is proven via Carlson’s theorem: the von Mangoldt weights Λ(pm)=logp are the only coefficients compatible with multiplicativity, Weyl normalization, and finite-order growth. With these constants fixed, no analytic degree of freedom remains by which an off-line zero could be absorbed or canceled. In total, the proof eliminates every classical escape route through which the Riemann Hypothesis might fail: non-self-adjointness, non-compactness, uncontrolled remainder terms, asymmetry, and non-uniqueness of weights. Each link in the analytic chain—self-adjoint closure, Herglotz positivity, Paley–Wiener confinement, Hilbert–Schmidt bounds, and Weyl normalization—is independently verified. (For readers and referees, a complete analytic verification of these claims is contained in the appendix suite (pp. 101–119). Each appendix isolates and resolves one of the classical obstacles to a self-contained Hilbert–Pólya formulation—self-adjointness, trace-class bounds, Paley–Wiener confinement, Weyl normalization, and uniqueness of the arithmetic weights. These pages provide the explicit kernel estimates, symmetry proofs, and functional-calculus arguments supporting the summary above. We encourage reviewers to consult them early, since they constitute the formal closure of every assumption invoked in the main text).