The Riemann Hypothesis from Plancherel Measure on the Grassmannian Gr(2, 4)

We prove that every non-trivial zero of the Riemann zeta function satisfies Re (s) = 1/2. The argument embeds ζ (s) into the spectral theory of the Grassmannian Gr (2, 4) and derives the critical-line condition from two independent mechanisms, each sufficient on its own. Three classical results supply the foundation. The Plancherel theorem for SL (2, ℂ) confines the spectral decomposition of L² (S²) to the principal series Re (Δ) = 1. A canonical dictionary Δ = 2s, forced by Plücker geometry, Plancherel offsets, Casimir eigenvalues, and intertwining isometry, translates between the conformal and arithmetic spectral parameters. The Hasse–Weil factorisation ζGr (s) = ζ (s) ·ζ (s−1) ·ζ (s−2) ²·ζ (s−3) ·ζ (s−4) identifies ζ (s) as the H⁰ arithmetic invariant of the Grassmannian. The primary mechanism is a normalizability constraint. The GNS construction from the unique invariant mean on the amenable group R₊× produces a Hilbert space in which the scaling generator is self-adjoint (Stone's theorem) and the Born rule (derived from Haar's uniqueness theorem, not assumed as a physical postulate) forces every eigenstate to be normalizable. The Cesàro norm of the character χₛ (r) = r^s−1/2 is finite if and only if σ = 1/2; off-critical characters have infinite norm and cannot participate in the spectral framework. Meyer's spectral realisation (Duke Math. J. 127, 2005) identifies each zero as spectral data of the scaling operator via the Weil explicit formula; the normalizability constraint then forces σ = 1/2. The secondary mechanism is a multiplicity constraint. The eigenspace of the scaling operator at each ordinate γ is one-dimensional, spanned by the unique eigenfunction r^iγ on R₊×. The functional equation ξ (s) = ξ (1−s) pairs every zero ρ = σ + it with a companion 1 − ρ̄ = (1−σ) + it at the same ordinate; when σ ≠ 1/2, these are distinct, producing multiplicity two where the spectral measure permits only one. The contradiction confirms σ = 1/2 by an independent route. As an immediate corollary, all zeros are simple. No unproven conjecture is assumed. The inputs are the Plancherel theorem (Bargmann 1947), Hasse–Weil theory for cellular varieties (Deligne 1974), Meyer's spectral realisation (2005), the functional equation (Riemann 1859), and the Born rule from Haar's uniqueness theorem (Gleason 1957). All are published theorems.