From Landscape to Proof: The Even-Dominance Route to the Riemann Hypothesis

This record is the three-part Landscape and audit-trail series for the even-dominance route to the Riemann Hypothesis. It documents the development path from foundations and obstruction analysis through failed routes, interval-arithmetic certificates, the Direct Frontier-Dominance route, and the v2.3 honest status correction that re-frames the result as a conditional reduction. The compact proof-only extraction is published separately as A Conditional Proof of the Riemann Hypothesis via Even Dominance of the Weil Quadratic Form (DOI: 10.5281/zenodo.20011910). Cite this record for the full research landscape, audit trail, excluded strategies, supporting certificates, and historical development of the route. For the shortest proof-core citation, use the separate proof-only record above. Three-part series: Part I Foundations (15p), Part II Even Dominance (52p), Part III Conclusio (20p). English + German (~174 pages total). Key results: Shift Parity Lemma: each prime individually favors even eigenfunctions 33 computer-assisted certificates (λ = 100 to 1,300,000) Leading-Mode Cancellation (c = 2 + √2) Direct Frontier-Dominance variational route, uniform in λ, with a single CAP base case at λ = 100 Two non-existence theorems NE-A (Aλ not PF∞) and NE-B (no universal commuting operator for N ≤ 15) ruling out the strongest algebraic alternatives OP2 (simplicity of ground state): interval-arithmetic certified at all 33 values NEW in v2.3: explicit conditional-reduction framing. The variational route establishes λmin(W+-W-) < 0; the named open conjecture "Asymptotic Variational Gap Conjecture" (lower bound on λ1-) is required to upgrade this to the eigenvalue ordering λ1+ < λ1- needed by Connes-vS Theorem 6.1. NEW in v2.3: Connes 2026 conditioning sharpened. Fact 6.4 (Fourier convergence of kΛ to Ξ with rate O(Λ-1/2-δ)) is established via the classical Slepian-Pollak prolate-spheroidal-wave theory (Connes 2026 §6.5, Ref. 47); only Even Dominance (this paper) and approximation quality kΛ ≈ ξΛ (companion Zookeeper paper) remain open in Connes' programme. 17 independent results, 11 explored alternatives CHANGELOG 2.3 Changelog (from 2.1): Honest status correction after external critical review (14 May 2026) and Connes 2026 §6.4-6.6 full-text study (15 May 2026). Status retracted from "unconditional A1-A8" to "conditional reduction". The v2.1 Direct Frontier-Dominance argument establishes λmin(W+-W-) < 0 (variational), but not λ1+ < λ1- (eigenvalue ordering). A separate lower bound on λ1- is required and is now formulated as the named open Asymptotic Variational Gap Conjecture (K0). Conditioning sharpened (Connes 2026 §6.4-6.6 fully studied). Fact 6.4 (Fourier convergence kΛ → Ξ with rate O(Λ-1/2-δ)) is, by Connes' own account, established via the classical Slepian-Pollak prolate-spheroidal-wave theory (Theorem 1 of Connes' Ref. 47) - not a conjecture. Connes-vS 2025 Theorem 6.1 (quadratic-form real-zero criterion) is proven. The two genuinely open points per Connes 2026 §6.6 are: (i) Even Dominance for QWΛ (the subject of this paper), (ii) approximation quality kΛ ≈ ξΛ (addressed in the companion Zookeeper paper, DOI 10.5281/zenodo.20151122). NE-A and NE-B retained as independent structural theorems. The non-existence theorems are unaffected by the status correction and remain available as standalone results ruling out the strongest algebraic routes. Reduction-Chain table in Part III updated with explicit variational / conditional / conditional reduction status labels reflecting the v2.3 honest framing. Companion-paper cross-references added: GeigerEvenDom2026 (the proof-only extraction record), Geiger2026zookeeper (CCM-route companion) in the bibliography and cross-references. Page counts: Part I (~15 pp), Part II (~52 pp), Part III (~20 pp), all compiled with zero undefined references. EN + DE. Former Changelogs you can watch in the former uploaded versions