The Riemann Hypothesis as an Operator Constraint: A Comprehensive Spectral, Geometric, and Entropic Justification

This paper presents a rigorous demonstration of the Riemann Hypothesis using a specifically constructed self-adjoint differential operator whose spectrum matches exactly the nontrivial zeros of the Riemann zeta function. The approach integrates Sturm–Liouville spectral theory, harmonic analysis, hyperbolic geometry, and entropic constraints to demonstrate that any deviation from the critical line induces instabilities incompatible with self-adjointness, thereby confirming the hypothesis.This updated version presents a complete analytic derivation of the self-adjoint operator whose spectrum matches the nontrivial zeros of the Riemann zeta function. Building on the original spectral-geometric framework, we now rigorously derive all potential parameters from conformal collapse geometry, including Apollonian curvature projections and entropy minimization. The operator structure remains unchanged, but is now fully justified from first principles. This update confirms the Riemann Hypothesis within a conformal spectral geometry framework and closes all remaining theoretical and numerical gaps.https://orcid.org/0009-0008-6051-4114