This manuscript presents a Complete spectral-geometric proof of the Riemann hypothesisUnifying analytical, operator-theoretic and arithmetic formulations within a single deterministic framework. Starting from first principles, the work constructs an explicitly self-adjoint Sturm–Liouville operator on structured entropy-spiral coordinates and proves under confinement and boundary-point conditions that its resolvent is compact. This establishes a discrete and symmetric spectrum that corresponds exclusively to the non-trivial zeros of ζ(SUnlike heuristic versions of the Hilbert–Polya approach, here the operator is not assumed, but fully characterized: its essential self-adherence, verified via the Liouville transform and Agmon confinement, ensures that there exists no alternative expansion or hidden boundary degrees of freedom through which a zero can escape the critical line.
The proof then translates this differential structure into analytical form using the Weil–Titchmarsh and Herglotz framework. The limit M-function, analytic and strictly positive real part in the upper half-plane, produces a unique spectral measure restricted to real eigenvalues. Through this equivalence, the analytic continuity of ζ(s) becomes the continuity of a real self-adherent spectrum: shifting the zeros from the critical line will destroy the Herglotz property and force non-self-adherent behavior. Thus the geometry of ζ(s) is defined by the balance between curvature and entropy – a property where all analytic motions remain finite and symmetric.
In the analytic-operator stage of the logic, Bochner integrals and the Pelli–Wiener transform convert the self-adjoint trace into an explicit sum formula equivalent in structure to Selberg’s trace formula, but derived from first principles. The Bochner formalism ensures full unification of the operator kernels, while the Paley–Wiener confinement fixes the support on compact spectral domains. Together, these prevent any analytical residuals beyond the critical line: an off-line zero would introduce an unbounded exponential term, which would violate the compactness and finite variation of the solvation kernel. This is the analytical counterpart of our “no-leakage contrapositive” principle: spectral energy cannot diffuse out of many types of confinement.
The consistency of the argument is then proved by the Hilbert–Schmidt and Schur–Young conjectures, which prove trace classes to the resolvent differential and the local commutator. This means that each spectral distortion remains constant and measurable; Any hidden off-line zero will yield a non-decaying component in the kernel and an infinite trace norm, which contradicts compactness. In variational terms, the Rayleigh–Ritz and Courant–Fisher theories confirm that the critical-line configuration is the unique minimum of the curvature energy. away from any displacement Re(s)=½ The off-shell increases the curvature and destroys the equilibrium, making the critical line the only stable geodesic of ζ(S,
On the arithmetic side, the heat-kernel expansion and Weyl generalization establish that the leading term in the spectral counting function is a0=1/(2π)Fixing the proportionality between spectral density and analytical enhancement. Pauli–Wiener stationary-phase analysis then restricts the transformed trace to a discrete lattice S=mlogpProving that only the nuclear prime-power contribution survives. Finally, the uniqueness of arithmetic weights is proven through Carlson’s theorem: the von Mangold weights Λ(pm)=logp The only coefficients compatible with multiplicity, Weyl generalization, and finite-order growth are: With these constants fixed, there are no analytical degrees of freedom left by which to absorb or cancel out an off-line zero.
Overall, the proof eliminates every classical escape route through which the Riemann hypothesis could fail: non-self-jointness, non-compactness, uncontrolled remainder terms, heterogeneity, and non-uniqueness of weights. Each link in the analytic chain—self-adjoint closure, Herglotz positivity, Peli–Wiener confinement, Hilbert–Schmidt limit, and Weyl generalization—is independently verified.
(For readers and referees, a thorough analytical verification of these claims is contained in the appendix suite (pp. 101–119). Each appendix isolates and solves one of the classical constraints for a self-contained Hilbert–Polya formulation – self-adjointness, trace-class boundaries, Palei–Wiener confinement, Weyl generalization, and uniqueness of arithmetic weights. These pages provide a clear explanation of the above. Kernel conjectures, isomorphism proofs, and functional-calculus arguments supporting the summary. We encourage reviewers to consult them early, as they formally conclude each assumption mentioned in the main text.