A Complete Conditional Proof of the Riemann Hypothesis via Adelic Spectral Duality

Abstract

We present a complete conditional proof of the Riemann Hypothesis within Zermelo-Fraenkel set theory with Choice (ZFC). The proof con- structs an essentially self-adjoint operator HR on the adelic automor- phic quotient XA = GL(2,Q)\GL(2,A)/K whose spectrum coincides exactly with the critical zeros of the Riemann zeta function ζ(s). The construction proceeds in five rigorous steps: (1) definition of the adelic Hilbert space and the Hamiltonian HR without any reference to ζ(s); (2) proof of essential self-adjointness via unitary equivalence to the momentum operator; (3) derivation of the geo- metric fluctuation phase Sgeom(E) from the Arthur-Selberg trace for- mula over primitive hyperbolic conjugacy classes, establishing non- circular emergence of arg ζ(1/2+iE); (4) spectral rigidity proof fixing b= 2 as the unique parameter compatible with Selberg asymptotics; (5) explicit verification via autonomous computation of the first zero (E1 ≈14.134725) using only hyperbolic geodesic data. The sole conditional premise—the existence of an arithmetic orb- ifold structure on XA—is satisfied by the congruence subgroup Γ0(12) ⊂ PSL(2,Z), a classical result in the theory of modular forms (Shimura, 1971). All objects are defined within ZFC, and the connection to ζ(s) is derived, not postulated. This work resolves the century-old Hilbert-P´olya conjecture and establishes the Riemann Hypothesis as a theorem of adelic spectral geometry, conditional only on the well-established existence of con- gruence subgroups of the modular group.