A Conditional Reduction of the Riemann Hypothesis to Even Dominance of the Weil Quadratic Form

This paper presents a conditional reduction and diagnostic programme toward the Riemann Hypothesis via even dominance of the Weil quadratic form. It extracts the proof-relevant chain from a three-part Landscape series while making the remaining mathematical gap explicit. (i) External preprint inputs: the Connes--van Suijlekom quadratic-form real-zero criterion and Connes' Fact 6. 4/Hurwitz bridge; (ii) Structural component: the Shift Parity Lemma and the Connes-vS single-shift asymmetry formula; (iii) Diagnostic evidence: 33 finite Landscape CAP values over 100 ≤ λ ≤ 1, 300, 000, currently not theorem-level because the odd-sector lower-bound construction remains open; (iv) Open internal bridge: the Asymptotic Variational Gap Conjecture, needed to lift variational frontier dominance to the eigenvalue ordering λ1+<λ1-. v1. 5 claim-level update (18 May 2026): the title and metadata have been reframed from "conditional proof" to "conditional reduction". The legacy frontier and tail-control sections are retained as diagnostic route documentation, not as completed theorem-level input in the corrected Connes--van Suijlekom Uₙ framework. The finite certificates remain numerical evidence pending recomputation and rigorous odd-sector lower-bound validation in that corrected framework. Changes in Version 1. 6 (May 2026) This maintenance version publishes the source-check corrections from 19 May 2026. Critical: no theorem-strengthening changes; the paper remains a conditional reduction and diagnostic research draft. Major: the Spectral Zookeeper self-citation was corrected from an older version DOI to the stable concept DOI 10. 5281/zenodo. 19673126. Minor: external metadata for Connes 2026, Connes--van Suijlekom 2025, Dusart 2010, Iwaniec--Kowalski and Kato was cross-checked against primary records. DE/EN: English and German PDFs were rebuilt from the synchronized v1. 6 source set; the combined PDF was regenerated from EN plus DE. The Riemann-Hypothesis implication is therefore conditional on the external Connes / Connes--van Suijlekom preprint inputs and on the internal Asymptotic Variational Gap Conjecture. This version is intended as a transparent research draft rather than a closed unconditional proof.