The Riemann Hypothesis: A Direct Proof via Even Dominance of the Weil Quadratic Form (DRAFT)

This paper extracts the proof core from a three-part Landscape series on the Riemann Hypothesis (Geiger 2026, Zenodo concept-DOI 10. 5281/zenodo. 19035640). The Landscape papers document the full research process, including failed routes, obstruction theorems, computational certificates, and alternative strategies. The present paper keeps only the proof-relevant chain. Following Connes' reduction of the Riemann Hypothesis to an even-dominance condition for the Weil quadratic form, we prove that the relevant ground state is simple and even along a sequence lambdaₙ -> infinity. The proof has four inputs: (i) Connes-van Suijlekom's real-zero criterion for quadratic forms (Theorem 6. 1 of Connes 2026, together with the Hurwitz bridge of Section 6. 4-6. 6) ; (ii) the Shift Parity Lemma, a finite trigonometric matrix theorem showing that every admissible prime shift has an intrinsic even-sector advantage; (iii) interval-arithmetic certificates for the finite range 100 <= lambda <= 1, 300, 000; and (iv) a frontier-dominance argument based on the Prime Number Theorem, Mertens-type estimates, and a common Rayleigh-vector bound for primes p with log p / log lambda near 1. Step (i) is taken as an external input from the Connes program; the proof of the Riemann Hypothesis presented here is therefore conditional on those two results until they are independently published. Steps (ii) - (iv) are self-contained and analytical, with the finite range and the asymptotic threshold meeting exactly at 1, 300, 000. DRAFT version. Subject to revisions. The excluded gauge, total-positivity, and universal-commutator routes are not used in the proof and are referred to the Landscape series.