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We prove the Riemann Hypothesis by establishing that the Riemann zeta function is spectrally isolated in the Weil quadratic form, separated from every nontrivial Dirichlet L-function by a permanent arithmetic gap. Zeta is the unique L-function with conductor 1: no character twist of the Euler product. Using the Lagarias identity to decompose the Weil form per character, we show that this minimality makes zeta the permanent spectral minimum of the even sector, isolated from all even twists by a fixed gap δ = 0.05518…, determined by the Legendre symbol modulo 5 and independent of the primorial level. Combined with the exact parity symmetry of the Weil form, this establishes the conditions of the Connes–van Suijlekom theorem: the spectral minimum of the Weil form is a simple, isolated eigenvalue with even eigenfunction. The theorem then forces all zeros of the eigenfunction's Fourier transform onto the real line. In the Weil context, these are the nontrivial zero ordinates of ζ, so the CvS conclusion is RH at each finite level. Hurwitz's theorem passes this to all nontrivial zeros. A second, independent proof via the Selberg trace formula for Γ₀(5), requiring neither the CvS theorem nor the Weil form machinery of the primary proof, confirms RH and extends to the full Generalised Riemann Hypothesis for all Dirichlet L-functions with no parity restriction. The proof resolves both steps identified by Connes (arXiv:2602.04022, Section 6.6) as missing from his programme: the twist ordering shows "the smallest eigenvalue of Q_Wλ is simple with even eigenvector" (Step 1), and the character decomposition bypasses Step 2: rather than proving that "kλ is a sufficiently good approximation of θx" directly, Davis–Kahan alignment and Connes' own prolate convergence show both converge to the same limit. Sixth paper in the Primorial Bridge Programme. Companion to The Kernel, The Prime Tree, The Bridge, The Chain, and the Vow.