From Landscape to Proof: The Even-Dominance Route to the Riemann Hypothesis

A three-part Landscape and audit-trail series for the even-dominance route to the Riemann Hypothesis via Connes' spectral program (arXiv:2602.04022). It documents the proof path, failed routes, obstruction analysis, computational certificates, and the transition from the broader research landscape to the separately published proof-only extraction.

Proof-only extraction: https://doi.org/10.5281/zenodo.20011910

This public repository is the curated reproducibility package for the Landscape series. It intentionally includes the paper sources, public scripts, and certificate outputs, while internal proof notebooks (BEWEISNOTIZ*.md, _proof-notes/, handoffs, local server-root captures, and credentials) remain private until a full project closeout permits release.

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

Current Status (v2.2, 2026-05-14)

The public paper package now reflects the v2.2 status correction. The certified range 100 <= lambda <= 1,300,000 remains rigorously established via separate interval-arithmetic bounds for lambda_1^+ and lambda_1^-, while the asymptotic step is reduced to the Asymptotic Variational Gap Conjecture. The Direct Frontier-Dominance argument remains part of the package as the asymptotic variational route, but it no longer by itself closes the eigenvalue inequality lambda_1^+ < lambda_1^-.

Paper Series (3 Parts, EN + DE)

Paper	File	Pages	Content
Part I	RH_I_Foundations	15	Foundations and Obstructions: thermodynamic landscape (R1-R9), dead ends (K1-K4), reorientation to Connes
Part II	RH_II_Even_Dominance	52	Main paper. Shift Parity Lemma, 33 CAP certificates, the M1'' variational framework, Leading-Mode Cancellation (c=2+sqrt(2)), Higher-Mode Decay (Lemma B), Resolvent Truncation (Lemma C), PNT Transfer, Euler-Maclaurin Proposition, Direct Frontier-Dominance, and the v2.2 status revision
Part III	RH_III_Conclusio	19	Synthesis of the proof architecture, revised v2.2 status table, explored alternatives (BI-1..11), independent results, and the remaining asymptotic gap problem

All papers are available in English and German (DE suffix). Combined English version: paper/RH_Complete_Series_EN.pdf (86 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	variational, v2.2
A7	Even dominance along lambda_n -> infinity	conditional (v2.2)
A8	RH	conditional reduction (v2.2)

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'' Variational Framework: The resolvent-damped comparison yields the asymptotic variational sign with explicit threshold lambda_0 = 442,413 (Dusart bound).

5. Current v2.2 status of Proposition A6:

   • Regime 1 (lambda in [100, 1.3M]): 33 CAP certificates give rigorous even dominance on the certified values via separate bounds for lambda_1^+ and lambda_1^-.
   • Regime 2 (asymptotic): M1'' + PNT Transfer + Lemma B + Lemma C identify the correct variational sign.
   • Remaining gap: asymptotic even dominance is reduced to the Asymptotic Variational Gap Conjecture for lambda_1^-.

6. v2.1 Direct Frontier-Dominance: independent asymptotic variational route using a common frontier Rayleigh vector, PNT partial summation, Mertens bounds, and finite CAP coverage through the explicit asymptotic threshold. It removes the earlier interpolation/PNT-constant caveats, but still requires a separate odd-sector lower bound to close the eigenvalue inequality.

7. 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
verify_lambda_star.py	Exhaustive check of the lambda* threshold logic
weighted_compactness_test.py	Weighted compactness test
weighted_compactness_server.py	Server version of compactness test

Results (results/)

File	Content
results/certificates/certificates.json	23 rigorous certificates (lambda 100-9201)
results/certificates/certificates_extended.json	29 certificates (lambda 10000-320000)
results/certificates/certificates_gap_closure.json	3 gap-closure certificates (700K, 1.05M, 1.3M)
results/certificates/euler_maclaurin_results.json	Euler-Maclaurin interval-arithmetic certification
results/certificates/largeN_results.json	Large-N certificate output
results/certificates/rigorous_results.json	Earlier rigorous certificate bundle
results/certificates/rigorous_v3_lam100.json	v3 lambda=100 certificate
results/certificates/rigorous_v3_results.json	v3 rigorous certificate summary
results/certificates/rigorous_v4_lam100.json	v4 lambda=100 certificate
results/certificates/rigorous_v4_lam200.json	v4 lambda=200 certificate
results/certificates/simplicity_certificates.json	OP2 simplicity certificates
results/gap_analysis/gap_monotone_results.json	Gap monotonicity analysis
results/gap_analysis/gap_monotone_v2_results.json	v2 gap monotonicity analysis
results/gap_analysis/hellmann_feynman_results.json	Hellmann-Feynman derivative analysis
results/gap_analysis/lipschitz_analysis.json	Gap-continuity Lipschitz analysis
results/gap_analysis/resolvent_analysis.json	Dense-grid resolvent energy analysis

Historical exploration code in scripts/_exploration/ and raw runtime logs in results/**/*.log stay local-only on purpose; the tracked files are the curated scripts and reproducible outputs needed for the public package.

Gemini Verification (scripts/gemini_verification/)

Independent full-Galerkin checks at lambda=100 and lambda=200 are archived with scripts, CSV outputs, and recovered server logs. The production run uses N=200, P_max=10000, and confirms 1229/1229 tested primes deepen the gap for both lambda values.

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

• 2.2 (2026-05-14): Status correction from unconditional closure to conditional reduction; public EN paper package synced to the variational-gap revision
• 2.1 (2026-04-30): Robust Direct Frontier-Dominance route, Gemini N=200 server-script archival, repo hygiene audit
• 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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