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. PAPERS Part I Foundations (14p), Part II Even Dominance (44p), Part III Conclusio (18p). English + German (152 pages total). KEY RESULTS Shift Parity Lemma: each prime individually favors even eigenfunctions 33 finite-range computer-assisted gap diagnostics (λ = 100 to 1, 300, 000), conditional on odd-sector tail validation Leading-Mode Cancellation (c = 2 + √2) Proposition A6: three-regime/direct-frontier variational bridge (conditional reduction) OP2 (simplicity of ground state): interval-arithmetic certified at all 33 values 17 independent results, 11 explored alternatives CHANGELOG 2. 4 Changelog (23 May 2026): Parallel programme addition and converged empirics. Critical: no theorem-strengthening changes; the record remains a conditional reduction and audit-trail series. New subsection in Part III (EN and DE): “Parallel Programme: Connes--vS Error Collapse and I-1 Normal-Family Reframe” (sec: parallel-route-b). Documents Route B: (a) A1 = CvS Theorem 5. 6 + err→0 ⇔ (i) + (ii) rigorously, (ii) ⇒RH rigorously; (b) I-1 Normal-Family Reframe (§E. 1–E. 18 Companion): wall (ii) = (ii-a) + (ii-c) via Montel/Vitali, (ii-c) absorbed into (ii-a) via parity and Hadamard order (under zero-coincidence assumption (Z) ) ; remaining open bridge: uniform a-priori strip bound (ii-a). Canonical criterion object §E. 15: ξ̂ (z) = (2/√L) ·sin (zL/2) ·F (z). New empirics table (tab: route-b-empirics, EN and DE): six converged λ-values (λ ∈ 3, 5, 7, 9, 11, 13): sup|ξ̂|ℜ ≈ 0. 86–0. 89; Aλ (0. 4) flat at ≈ 1. 004; Bλ/λ0. 4 plateau at 0. 68 (λ≥9). λ=11 and λ=13 bit-identical across two independent dps-runs. DOI correction for Geiger2026evendom: updated from 10. 5281/zenodo. 20011910 (older version record) to 10. 5281/zenodo. 20291994 (v1. 6 record). DE bibitem added. Note: after EvenDom v1. 7 push, update again to the new v1. 7 record DOI (TBD nach EvenDom-v1. 7-Push). Version strings: four occurrences of “v2. 3 status” updated to “v2. 4 status” in Part II EN and DE. Parts I (EN+DE) unchanged. 2. 3 Changelog (15 May 2026): Status correction and proof/paper synchronization. Claim level corrected: v2. 1's "unconditional A1--A8" claim is withdrawn. The record is now a conditional reduction. Mathematical correction: the Direct Frontier-Dominance argument proves the variational sign _ (W^+-W^-) <0, not by itself the eigenvalue ordering ₁^+<₁^-. Finite CAP guardrail: the 33 finite-range values are retained as strong numerical evidence / conditional finite bridge until the odd-sector tail lower bound is independently validated. Conditioning clarified: Connes 2026 Fact 6. 4 is treated as established via Slepian--Pollak theory; the internal open RH step is asymptotic even dominance / odd-sector lower-bound closure. 2. 1 Changelog (from 2. 0, historical; superseded by v2. 3 status correction): Historical v2. 1 claim, superseded by v2. 3: the Riemann Hypothesis reduction was then stated as unconditional. Direct Frontier-Dominance addition (Part II, Proposition prop: direct-frontier, Section sec: direct-frontier). The argument remains valuable as a proof of the asymptotic variational sign. Its original v2. 1 interpretation as an independent, uniform-in-λ proof of the eigenvalue-ordering Proposition A6 is withdrawn. Superseded caveat status. The direct argument does bypass the old v2. 0 interpolation/PNT-transfer caveats for the variational statement, but it is not unconditional for the eigenvalue-ordering claim on all λ ≥ 100. Proof architecture A1--A8 corrected in v2. 3. A7 (Even Dominance for all λ ≥ lambda₀) and A8 (RH) are not proved unconditionally by v2. 1; they remain conditional on the odd-sector lower-bound / AVG closure. Numerical evidence across 9, 705 prime additions (lambda ∈ 100, 200, 500, 1000, 2000, 5000, 10000, 20000, 50000): zero exceptions to the per-prime Rayleigh negativity, consistent with the lambda^-1/2 shrinking of per-prime contributions against a sqrt (lambda) -growing aggregate. Paper II (English and German), Paper III (English and German) updated with the Direct Frontier-Dominance Proof, revised abstracts, OP1-Remark, Reduction-Chain status tables, Gap-Closure-Path conclusion, Summary tables, and Conclusion statements. Both v2. 0 caveats are retained as historical remarks documenting the three-regime proof's v2. 0 status. 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 lambda-proof three-regime bridge. 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 Reζ'/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. 3 presents the program as a conditional reduction of RH to explicit analytic refinements: the finite-range diagnostics are conditional on the odd-sector lower-bound validation, and the asymptotic variational argument still needs a quantitative lower bound for λ1-. The Direct Frontier argument is retained as a variational result, not as an unconditional eigenvalue-ordering proof. 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=200 Galerkin discretization, primes up to Pₘax = 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/rhᵥ2ᵢmpactₐnalyses/`: (i) FSTIMPACTREPORT. md (framework/CRM/follow-on proofs -- additive updates only, no destructive impact), (ii) POTENTIALRHANALYSE. 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 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 t