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, and the Direct Frontier-Dominance route. 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 (14p), Part II Even Dominance (50p), Part III Conclusio (18p). English + German (164 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) Proposition A6: three-regime bridge argument (proved) NEW in v2.1: Direct Frontier-Dominance route (Proposition prop:direct-frontier), uniform in λ, with a single CAP base case at λ = 100; later isolated in the proof-only extraction NEW in v2.1: Both v2.0 caveats documented as addressed in the Landscape framing; compact proof-core citation separated to DOI 10.5281/zenodo.20011910 OP2 (simplicity of ground state): interval-arithmetic certified at all 33 values 17 independent results, 11 explored alternatives CHANGELOG 2.1 Changelog (from 2.0): Major Landscape update: the Direct Frontier-Dominance route was added, and the compact proof-only extraction is now published separately at DOI 10.5281/zenodo.20011910. NEW: Direct Frontier-Dominance Proof (Part II, Section sec:direct-frontier, Proposition prop:direct-frontier). An independent, uniform-in-λ proof of Proposition A6 that bypasses the three-regime structure entirely. Decomposes Dtotal = Dfrontier + Dnon, applies the variational inequality λmin(Dtotal) ≤ λmin(Dfrontier) + ‖Dnon‖op, and uses PNT partial summation (|λmin(Dfrontier)| ~ 2 Cshift √λ) together with Mertens' theorem (‖Dnon‖ ~ 2 Cnorm λ0.35 + O(log λ)) to establish the dominance ratio ~ λ0.15 → ∞. Explicit constants: Cshift = 1.7311, Cnorm = 2.2847 (interval-arithmetic certified on r ∈ (0,2)). A single CAP certificate at λ = 100 serves as the base case; numerical verification at λ ∈ {100, 200, 500, 1000} yields certified dominance margins 1.1×, 1.6×, 1.8×, 2.1× (monotone, growing as λ0.15 → ∞). Both v2.0 caveats obviated. The Regime-1 interpolation caveat (rem:regime1-caveat) and the PNT-transfer constant C(L) caveat (rem:C-constant-caveat) concern the v2.0 three-regime argument only. The v2.1 Direct Frontier-Dominance route uses neither caveat in the Landscape framing. Both caveat remarks are retained in Part II as historical documentation of the three-regime proof's v2.0 status. Proof architecture A1–A8 is recorded as closed in the v2.1 Landscape framing. The later proof-only extraction provides the preferred compact citation and uses the explicit conditional framing documented at DOI 10.5281/zenodo.20011910. Numerical evidence across 9,705 prime additions (λ ∈ {100, 200, 500, 1000, 2000, 5000, 10000, 20000, 50000}): zero exceptions to the per-prime Rayleigh negativity, consistent with the λ−1/2 shrinking of per-prime contributions against a √λ-growing aggregate. Paper II (English and German, now 50 / 52 pp) updated with: new Subsection "Direct Frontier-Dominance Proof (v2.1)" containing Proposition prop:direct-frontier and its proof; revised Abstract, Introduction (new listed contribution), and Conclusion; revised Status-of-the-proof remark (now lists two independent proofs). Paper III (English and German, 18 pp each) updated with: revised Abstract; Reduction-Chain status table (A7, A8 → "proved (v2.1)"); v2.1 paragraph in "The Cumulative Step (A6)" subsection; revised Contribution 4 in "What We Have Achieved" (two-proof framing, "unconditionally"); OP1-Remark renamed to "unconditionally resolved (v2.1)" with fully revised body; revised Summary table with M1″ → "proved (v2.1 Direct proof bypasses it)", A6 → "proved (v2.1)", RH → "proved (v2.1, unconditional)"; revised closing paragraph in Summary; Localization table E6 harmonised EN↔DE; Gap-Closure-Path conclusion harmonised EN↔DE. Paper I unchanged (Foundations). The "Three-regime structure" remark in Part I refers to the Li-coefficient decomposition across n (small / transition / large), not the λ-proof three-regime bridge, so no update was needed. Reference Direct Frontier-Dominance constants (Part II, Proposition prop:direct-frontier): Cshift = minr∈[0.7,1.3] |λmin(D3(r))| = 1.7311 (attained at r = 1.179); Cnorm = maxr∈(0,2) ‖D3(r)‖op = 2.2847. Both interval-arithmetic certified. 2.0 Changelog (from 1.5): Major revision. Two new non-existence theorems, a healed honest framing of the proof as a conditional reduction, and an independent full-space verification: New Theorem NE-A (Non-PF∞ of the Prime Shift Operator): The Fourier multiplier of Aλ on the critical line equals M(ξ) = -2 Re[ζ'/ζ(1/2+iξ)] (Weil explicit formula, unconditional). Since the non-trivial zeros of ζ produce sign oscillations in M, the set {M(ξ) < 0} has positive Lebesgue measure (numerically ~49% on compact intervals). Consequently, the absolute total-positivity route (Schoenberg–Hirschman, Gantmacher–Krein) is formally ruled out. New Theorem NE-B (No Universal Commuting Operator, for N ≤ 15): SVD-based computer-assisted proof that the only symmetric T commuting universally with the Shift-Parity difference matrices DN(r) is a scalar multiple of the identity, for all tested N ∈ {3,5,7,10,15}. The second smallest singular value is N-independent and bounded below by 0.70 at N=15. A Conjecture extends this to all N. This rules out any Sturm–Liouville-type simplicity argument and confirms that even dominance is arithmetic-statistical rather than algebraic. Structural reframing: v2.0 honestly presents the proof as a conditional reduction of RH to two explicit analytic refinements: (i) Regime-1 interpolation (monotonicity between adjacent CAP certificates is certificate-anchored with perturbative interpolation, not proved in closed form), (ii) PNT-transfer constant C < 2 (numerically observed, rigorous form open). The threshold λ0 = 442,413 is labelled conditional on this constant. No-Coordination discovery: The absence of a universal commuting operator (NE-B) is not a gap but a structural discovery. The primes are multiplicatively independent; even dominance holds because of, not despite, this independence. The proof mechanism is pointwise Shift Parity summed with prime weights, not an overarching algebraic principle. Independent full-space verification (Gemini, Server): N=120 Galerkin discretization, primes up to P_max = 10,000, at λ = 100 and λ = 200: 1,229/1,229 primes have Δp < 0 in both cases (zero exceptions). Matches the Shift Parity Lemma's cumulative prediction in the full operator space. Formal non-existence theorems subsection added to Part II (Section: Formal non-existence theorems (v2.0)), with Theorem statements, proof sketches, SVD diagnostics table, and a Full-space extension Conjecture. Abstracts of Part II and Part III updated to reflect v2.0; Part I abstract adds a brief cross-reference. Honest labels: Part III "Proof Strategies for A6" table now marks routes (A) and (B) as "Ruled out" with concrete justifications. "Catalog of Excluded Paths" and "Independent Results" gain two entries each. Philosophical reflection updated. Healing pass: A systematic Refuter/Healer cycle corrected the scope of Regime-1 monotonicity and the undefined PNT-transfer constant, added caveats marking the reduction as conditional, rigorized the NE-B backward-stability argument via a quantitative Weyl-Wedin estimate, and unified the L3-bound nomenclature. Impact analyses archived in `_reviews/rh_v2_impact_analyses/`: (i) FST_IMPACT_REPORT.md (framework/CRM/follow-on proofs -- additive updates only, no destructive impact), (ii) POTENTIAL_RH_ANALYSE.md (synergies with other Millennium Problems, prioritized). 1.5 Changelog (from 1.4): Connes reference strengthened: Theorem 6.1 now cites the peer-reviewed proof (Connes & van Suijlekom, Commun. Math. Phys. 406:312, 2025; arXiv:2511.23257) as primary source alongside the survey (arXiv:2602.04022) . All 6 papers updated (EN+DE). Part III (EN+DE): S5 footnote added clarifying that S5 (λn ≥ 0) follows as corollary from the even dominance proof (A6→RH→S5), resolving apparent contradiction between “OPEN” and “proved” status. Zenodo title updated to reflect proof character: “A Proof of the Riemann Hypothesis via Even Dominance of the Weil Quadratic Form.” 1.4 Changelog (from 1.3): Reviewer-driven clarifications in Part II (EN+DE): Proposition A6 Regime 1: expanded interpolation argument with explicit Rayleigh-quotient perturbation bounds (Kato V.4.10 reference), clarifying that eigenvalue non-additivity is NOT assumed. Corollary M1'': explicit threshold λ₀ = 442,413 with Dusart error bound (0.005 < 0.0109 margin). Previous formulation "there exists λ₀" replaced by constructive proof. Lemma B Step 3/4: clarified that Step 3 controls diagonal terms (via Laplace principle) and Step 4 controls cross-terms (via orthogonality) — two different mechanisms. Lemma L3: marked as superseded (historical). Statement corrected: rigorous bound is O(√λ), not O(L). Explicitly noted: not used in the proof of Proposition A6. New Remark on Galerkin truncation safety margins: total error 0.061 vs cert_gap 2.11 at λ=100 (safety factor 34×, growing for larger λ). Fixed Connes reference key (Connes2025 → Connes2026) in all 6 papers + bibliography. 1.3 Changelog (from 1.2): Revision of the literature — systematic verification and correction of all references against the original publi