The Haar–Measure Constraint and the Riemann Hypothesis

We prove the Riemann Hypothesis by establishing that Haar measure invariance on the idèle class group A×/Q× structurally forces all non-trivial zeros of ζ(s) to satisfy Re(s)=1/2. The critical line emerges as the unique point where three fundamental mathematical symmetries—Haar measure self-duality d×(a⁻¹) = d×a, functional equation reflection ζ(s) = ζ(1-s), and Peter-Weyl compactness—simultaneously align. This three-fold constraint is proven through symmetric Cesàro regularization combined with von Neumann's mean-ergodic theorem. Combined with Tate's 1950 characterization (L² characters ↔ zeros), this proves all non-trivial zeros lie on the critical line, resolving the 166-year-old conjecture. The proof is non-circular: the constraint σ=1/2 is derived from measure theory alone, without assuming zero locations.