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

Abstract

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-. 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. CHANGELOG 2.1 Changelog (15 August 2026): Maintenance release (strict-review corrections and reference updates); no theorem-strengthening changes, the record remains a conditional reduction. Strict-review corrections (June 2026) now live: the Route-B observable Aλ(δ) is stated as the global strip amplification ratio (the earlier polynomial-weighted real-axis formula is neutralised as a transcription deviation); the bias comparison with AΞ is marked as internal diagnostic and falsification test, not a theorem-strength RH equivalence and not a substitute for the open bound (ii-a); the German κ-passage is synchronised with the English text (the outdated κ∞ claim is removed); the Sylvester-inertia formulation is restricted to finite-section inertia; CAP/Cauchy constant labels are downgraded to legacy diagnostics. Far-ring imbalance guardrail: the far-ring imbalance ledger (QA2_FAR_RING_IMBALANCE_BOUND) is mirrored into the paper as a diagnostic-only instrument; no DZT2/MS2/A7/RH claim upgrade. Companion references updated: landscape/atlas companion cited at v3.2 (10.5281/zenodo.21952620), Zookeeper companion at v1.6 (10.5281/zenodo.21953305). Status: conditional reduction unchanged; no new proof claim. 2.0 Changelog (14 June 2026): Maintenance update: PDF design, combined-PDF link targets, and bibliographic metadata; no theorem-strengthening changes, conditional-reduction status unchanged. PDF design: visible title-page date/version markers were simplified to June 2026, the German bibliography/disclosure ending was tightened, and the combined PDF was regenerated. Combined PDF links: English and German hyperref destination prefixes were made effective, eliminating cross-half internal jumps in the combined PDF. Bibliography metadata: the Spectral Zookeeper self-citation was updated to the current title and live v1.5 record DOI; Landscape part notes were aligned with live v3.1, with protected capitalization for key proper names. DE/EN: English and German PDFs were rebuilt synchronously; German text extraction confirms real UTF-8 umlauts and no mojibake. Status: no new proof claim; the RH implication remains conditional on the external Connes / Connes--van Suijlekom inputs and the Asymptotic Variational Gap Conjecture. 1.9 Changelog (31 May 2026): Additive diagnostic update from a cross-route research phase; no theorem-strengthening changes, conditional-reduction status unchanged. New subsection "Update (31 May 2026): Cross-Route Diagnostic Phase — Unconditional Structural Results, the Λ=0 Lens, and Why the Route Lives" (EN and DE), at the end of the asymptotic lower-bound section. Two unconditional structural results strengthening the route: a concentration lemma (no fixed finite prime set controls a positive share of the prime mass — the route has no few-prime/local key) and a finite Sylvester-inertia certificate (ν−=0 at λ=100), lifting the finite stage from heuristic tail-extrapolation to rigorous finite-section positivity. One sub-method closed, one re-typed: the Toeplitz-symbol lower-bound sub-method is closed (Asin has no well-defined symbol); the monotone-continuation bridge is A7-equivalent (re-typed, not closed). The Λ=0 criticality lens (RH ⇔ Λ=0): heuristic, not an impossibility theorem (the even-dominance margin is not shown to equal a de Bruijn–Newman statement). The route is not refuted — the asymptotic variational gap remains a well-posed open problem and the route lives; it is strategically deprioritised relative to exact-structural routes. Status: no new proof claim; the RH implication remains conditional on the external Connes / Connes–van Suijlekom inputs and the Asymptotic Variational Gap Conjecture. 1.8 Changelog (26 May 2026): Guardrail correction after the v1.7 parallel-route release; no theorem-strengthening changes. Versioning: v1.7 already exists on Zenodo (Record 20358727, 23 May 2026). This push is v1.8 and contains the 26 May guardrail corrections. Route B strip update: the λ=20 strip run gives κ(20)=1.527×10-3; the conservative six-point saturation range is κ∞∈[1.6,1.8]×10-3. Theorem B-1 remains conditional on uniform log-curvature control. GF1 correction: λ=5,7 support the boundary-moment benchmark sLI∼(64π2/3)λ/(NL), while λ=3 is a low-bandwidth outlier. This is not a uniform all-λ theorem and not a DZT2/MS2/RH closure. Claim wording: abstract wording changed from a conditional "proof" phrasing to a conditional route/reduction phrasing; EN/DE date headers now read v1.8. Changes in Version 1.7 (May 2026) This version adds a parallel proof route and converged empirical data. Critical: no theorem-strengthening changes; the paper remains a conditional reduction and diagnostic research draft. The RH implication stays conditional on the external Connes / Connes--van Suijlekom preprint inputs and the internal Asymptotic Variational Gap Conjecture. New: parallel route subsection (sec:route-b-parallel, EN and DE) — "Parallel Route via Connes-vS Err-Collapse and I-1 Normal-Family Reframe". Documents Route B: wall (ii) = (ii-a)+(ii-c) via Montel-Vitali; (ii-c) absorbed into (ii-a) via parity and Hadamard order (modulo zero-coincidence assumption (Z)). Remaining open blocker: uniform a-priori strip bound (ii-a) from QW_N-coercivity. New: seven-point converged empirics table (λ ∈ {3,5,7,9,11,13,15}): sup|ξ̂|ℜ ≈ 0.858 → 0.892; Aλ(0.4) flat in the range 1.003–1.004; Bλ/λ0.4 plateau at 0.68 (λ≥9). λ=11,13 bit-identical across two independent dps-runs; λ=15 bit-identical at dps=640 and dps=800. Internal note (not in paper): §E.16/E.17 retraction (dps=30 not convergent at near-degenerate even-sector minimum) documented in proof-notes only; only the converged §E.18 values are reliable. DE/EN: English and German PDFs rebuilt from synchronized v1.7 source set; combined PDF regenerated from EN plus DE. 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. 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 U_n framework. The finite certificates remain numerical evidence pending recomputation and rigorous odd-sector lower-bound validation in that corrected framework.