A Conditional Proof of the Riemann Hypothesis via Even Dominance of the Weil Quadratic Form

This paper presents a conditional proof of the Riemann Hypothesis, extracted from a three-part Landscape series. 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 λ_n → ∞. 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 ≤ λ ≤ 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 λ 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 at Λ* ≤ 175,000 (improved from 1,300,000 by exhaustive step-1 prime enumeration over [171,329, 182,000]). The excluded gauge, total-positivity, and universal-commutator routes are not used in this proof and are referred to the Landscape series. v1.2 additions (2026-05-03): (a) Hurwitz-bridge transparency remark (Paley–Wiener class PWL, Montel compactness, L²-rate control); (b) Λ* ≤ 175,000 via exhaustive step-1 verification over [171,329, 182,000] and step-1000 margin argument over [182,000, 300,000]; (c) German translation (13 pp); (d) combined EN+DE edition (25 pp); (e) verify_lambda_star.py added to companion repository.