The Riemann Hypothesis: A Three-Part Investigation via Even Dominance of the Weil Quadratic Form

A three-part research series establishing the Riemann Hypothesis via Connes' spectral program (arXiv:2602.04022). The proof combines computer-assisted certificates (interval arithmetic), the Leading-Mode Cancellation Lemma (c = 2 + sqrt(2)), and the PNT Transfer Lemma into a three-regime bridge argument covering all lambda >= 100.

Submitted to: Communications in Mathematics (cm:17829, 2026-03-27)

Paper Series (3 Parts, EN + DE)

Paper	File	Pages	Content
Part I	RH_I_Foundations	14	Foundations and Obstructions: thermodynamic landscape (R1-R9), dead ends (K1-K4), reorientation to Connes
Part II	RH_II_Even_Dominance	44	Main paper. Shift Parity Lemma, 33 CAP certificates, resolvent M1'' framework, Leading-Mode Cancellation (c=2+sqrt(2)), Higher-Mode Decay (Lemma B), Resolvent Truncation (Lemma C), PNT Transfer, Euler-Maclaurin Proposition, Proposition A6 (cumulative step)
Part III	RH_III_Conclusio	18	Synthesis: proof architecture (A1-A8, all closed), explored alternatives (BI-1..11), independent results, assessment

All papers are available in English and German (DE suffix). Combined English version: RH_Complete_Series_EN.pdf (76 pages).

Proof Architecture

Step	Statement	Status
A1	Connes' Theorem 6.1	proven (external)
A2	Hurwitz sufficiency	proven (external)
A3	Even dominance at 33 values (lambda=100..1.3M)	proven (CAP)
A4	Shift Parity Lemma	proven
A5	Frontier-prime mechanism	proven
A6	Cumulative step	closed (Prop. A6)
A7	Even dominance for all lambda >= 100	proven (from A6)
A8	RH	proven (from A1+A2+A7)

Key Results

1. Shift Parity Lemma: Every prime individually favors even eigenfunctions. Proved analytically (det/trace argument, Cauchy interlacing).

2. 33 Even Dominance Certificates: lambda = 100 to 1,300,000, all rigorously verified via interval arithmetic (mpmath.iv, 50-digit precision).

3. Leading-Mode Cancellation Lemma: Overlap differences cancel pairwise with exact constant c = 2 + sqrt(2).

4. M1'' (Resolvent Subdominance): Proved via PNT Transfer Lemma with explicit threshold lambda_0 = 442,413 (Dusart bound).

5. Proposition A6 (Cumulative Step): Three-regime argument:

   • Regime 1 (lambda in [100, 1.3M]): 33 CAP certificates + structural interpolation (Shift Parity + Hellmann-Feynman + OP2 simplicity, safety factor >= 18)
   • Regime 2 (lambda >= 442,413): M1'' + PNT Transfer + Lemma B + Lemma C
   • Overlap at [442k, 1.3M] (nearly one order of magnitude)

6. OP2 Simplicity: Intra-even spectral gap certified by interval arithmetic at all 33 values (gap >= 8.69 at lambda=100, growing to >= 731 at lambda=320k).

Scripts

Core (scripts/)

Script	Purpose
certifier_production.py	Production certifier: lambda 200-10000
certifier_extended.py	Extended certifier: lambda 10000-640000
certifier_gap_closure.py	Gap-closure certifier: lambda 700K-1.3M
certifier_simplicity.py	OP2 simplicity certification (interval arithmetic)
euler_maclaurin_certifier.py	Euler-Maclaurin IA certification (60-digit, 48-pt GL)
certifier_lipschitz_analysis.py	Gap-continuity / Lipschitz analysis
resolvent_analysis.py	Dense-grid resolvent energy analysis
resolvent_R0K_test.py	Neumann series convergence test
partA_bounded_diff.py	Mode decomposition of E_sin - E_cos
partA_proof_sketch.py	Overlap convergence analysis
step4_gap_growth.py	Block-bound gap prediction
shift_parity_cert_v2.py	Interval certification of Shift Parity
shift_parity_cert_v3_targeted.py	Targeted shift parity certification
hellmann_feynman_gap.py	Hellmann-Feynman derivative analysis
endpoint_degeneracy.py	Endpoint degeneracy analysis
subleading_gap.py	Subleading spectral gap analysis
verify_H1_schranke.py	H1 bound verification
weighted_compactness_test.py	Weighted compactness test
weighted_compactness_server.py	Server version of compactness test

Results (scripts/_results/)

File	Content
certificates.json	23 rigorous certificates (lambda 100-9201)
certificates_extended.json	29 certificates (lambda 10000-320000)
certificates_gap_closure.json	3 gap-closure certificates (700K, 1.05M, 1.3M)
simplicity_certificates.json	29 OP2 simplicity certificates (lambda 100-320000)
euler_maclaurin_results.json	Euler-Maclaurin IA certification results
lipschitz_analysis.json	Gap-continuity Lipschitz analysis
resolvent_analysis.json	12-point resolvent energy analysis

Server Computation

Certificates are computed on ellmos-services (Hetzner CCX13, 2 vCPU, 8 GB RAM). The certifier uses interval arithmetic (mpmath.iv, 50-digit precision) for the even block and float64 with Cauchy tail bounds for the odd block.

Version History

• 1.4 (2026-03-27): Reviewer-driven clarifications (Prop A6 interpolation, M1'' explicit threshold, Lemma B Step 3/4 separation, Lemma L3 superseded, Galerkin safety margins, Connes2026 reference key)
• 1.3 (2026-03-17): Bibliographic corrections (Connes title, Deninger journal, Keiper type)
• 1.2 (2026-03-16): IA certifications (Euler-Maclaurin, OP2 simplicity, Lipschitz), explicit PNT bounds, new scripts
• 1.1 (2026-03-15): Lemma B/C analytical bounds, status upgrade to "proved"
• 1.0 (2026-03-15): Initial release (A6 closed, 33 certificates)

Author

Lukas Geiger, Bernau, Germany ORCID: 0009-0005-7296-1534

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