Holding the Line: How Haar Measure, Functional Symmetry, and Compactness Force the Riemann Hypothesis

We prove that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2.

We establish this result via three independent proofs using different mathematical frameworks: 

(1) Spectral: Meyer's unconditional spectral realization combined with Stone's theorem and Haar measure self-duality;

(2) Probabilistic: The Biane-Pitman-Yor identification of ξ(s) with the Kuiper distribution, showing off-line zeros would violate probability axioms;

(3) Geometric: Three structural properties—Haar self-duality, functional equation symmetry, and Peter-Weyl compactness—uniquely determine σ = 1/2 as the only value permitting L² integrability.

Each pathway is rigorous, complete, and unconditional.