A Conditional Reduction of the Riemann Hypothesis to Even Dominance of the Weil Quadratic Form
Year: 2026
Type: preprint
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. 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.
Author: Geiger, Lukas
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162208301