Null-Metric Surfaces and the Riemann Hypothesis: A Geometric Reformulation via Connes' Noncommutative Geometry

Following the negative result established in Part 2 of this trilogy (the Phase-Volume Obstruction, DOI: 10.5281/zenodo.20672908), which ruled out local resonator models on classical Riemann surfaces, we transition to the global noncommutative geometry of the adèle class space C_Q = A_Q / Q*. We introduce the concept of a null-metric surface — the geometric locus where the spectral distance between a character and its conjugate vanishes — and show that it coincides with the critical line Re(s) = 1/2. We prove that the Riemann Hypothesis is mathematically equivalent to the assertion that the spectrum of the scaling operator on the reduced Hilbert space L²₀(C_Q) is purely discrete. This isolates the Spectral Barrier (Condition C5') as the single remaining obstacle in Connes' program, translating the analytic problem of zero distribution into a spectral question about the absence of continuous spectrum. This paper is Part 3 of a trilogy:• Part 1: Architectural Isomorphism (DOI: 10.5281/zenodo.20488420)• Part 2: Phase-Volume Obstruction (DOI: 10.5281/zenodo.20672908)• Part 3: Null-Metric Surfaces (this paper)