From Landscape to Atlas: Multi-Route Cartography of an Ongoing Expedition Toward the Riemann Hypothesis

Research status: public reproducibility package and audit-trail atlas. This repository documents conditional reductions, explored routes, failed paths, and computer-assisted certificates. It does not claim an unconditional proof of the Riemann Hypothesis.

A five-part Landscape/Atlas and audit-trail series mapping multiple routes toward the Riemann Hypothesis via Connes' spectral program (arXiv:2602.04022). It documents the explored routes, failed paths, obstruction analysis, computational certificates, and the transition from the broader research landscape to the separately published proof-only extraction (Even Dominance).

Discovery and Status Boundaries

Use this repository when looking for a reproducible Riemann Hypothesis research atlas, conditional-reduction audit trail, Zenodo-linked proof archive, or public certificate package for the Even-Dominance programme. It is intentionally positioned as a research record and verification surface: the public files are the paper sources, selected scripts, certificate outputs, citation metadata, and machine-readable context needed to inspect the Landscape/Atlas series.

For search and indexing, the canonical phrases are:

• Riemann Hypothesis research atlas
• Even Dominance conditional reduction
• Connes spectral program reproducibility package
• Zenodo Riemann Hypothesis certificate archive
• Weil Quadratic Form audit trail

Status boundary: this repository should be cited as a conditional research programme and reproducibility package, not as an unconditional proof claim.

Citation and Machine-Readable Context

• Landscape/Atlas Concept DOI: 10.5281/zenodo.19035640
• Latest verified Landscape/Atlas record: 10.5281/zenodo.20479302 (v3.1)
• Companion Even-Dominance Concept DOI: 10.5281/zenodo.19764771
• Latest verified Even-Dominance record: 10.5281/zenodo.20479145 (v1.9)
• Citation metadata: CITATION.cff
• LLM/crawler context: llms.txt

Proof-only extraction (Even Dominance, Concept-DOI): https://doi.org/10.5281/zenodo.19764771

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 (v3.1, 2026-05-31 sync)

The series has been restructured from the former three-part Trilogy into a five-part Atlas (Parts I-V; structural reorganization only -- the conditional reduction status is unchanged). The finite range 100 <= lambda <= 1,300,000 remains strong numerical evidence and a conditional finite bridge; theorem-level finite even dominance still depends on validating the odd-sector tail lower bound for lambda_1^-. The asymptotic step is reduced to the Asymptotic Variational Gap Conjecture.

Parts III-IV now carry the spectral programme explicitly: Part III documents the live spectral routes (Branch B: I-1 normal-family / Hadamard-strip, wall (ii-a); Branch Z: CCM microcluster closure, MS2), while Part IV collects the cross-route synthesis, dormant/external routes, and the excluded-paths catalog (uniform-gap eigenvalue ordering, naive strip localisation, edge-profile factorisation, literal MS1 cluster splitting, retracted low-precision strip readings). The common obstruction pattern is the near-degenerate minimum-eigenvector cluster of QW_N, which explains why productive routes have to work with cluster / quotient reformulations and growth-adapted normalisations rather than with uniform-gap or thin-localisation heuristics.

Paper Series (5 Parts, EN + DE)

Paper	File	Pages (EN)	Content
Part I	RH_I_Foundations	15	Foundations, obstructions, and reorientation: thermodynamic landscape (R1-R9), dead ends (K1-K4), reorientation to Connes
Part II	RH_II_Even_Dominance	55	Main route (A). Shift Parity Lemma, 33 finite-range diagnostics, 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
Part III	RH_III_SpectralPaths	13	Spectral routes (B, Z). Branch B (I-1 normal-family / Hadamard-strip, wall (ii-a)); Branch Z (CCM microcluster closure, MS2); spectral dead ends
Part IV	RH_IV_CrossRoute	13	Cross-route synthesis and remaining paths: inter-branch diagnostics, the sharpened common wall, dormant routes (D-CCM/Twist/PW, P-M, M), external lines, excluded-paths catalog, methodological dead ends
Part V	RH_V_Conclusio	7	Conclusio: what is proven, what is excluded, and what remains

All papers are available in English and German (DE suffix). Combined English version: paper/RH_Complete_Series_EN.pdf (103 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)	numerical evidence / conditional finite bridge
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 finite-range gap diagnostics: lambda = 100 to 1,300,000, with interval-arithmetic even upper bounds and a currently conditional odd lower-bound component.

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 v2.3 status of Proposition A6:

   • Regime 1 (lambda in [100, 1.3M]): 33 CAP diagnostics give strong finite-range evidence; theorem-level even dominance on the certified values requires validation of the odd-sector tail lower bound.
   • 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 diagnostic 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 were computed on a dedicated cloud instance (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

• 3.1 (2026-05-31): Repo sync of the Atlas pentalogy; companion Even-Dominance paper updated to v1.9 (10.5281/zenodo.20479145); Part I rebuilt for the five-part framing; combined EN PDF regenerated (103 pages); internal strategy/QA files kept private via .gitignore
• 3.0 (2026-05-26): Restructured the three-part Trilogy ("From Landscape to Proof") into the five-part Atlas ("From Landscape to Atlas"): Part III split into Spectral Routes (III), Cross-Route Synthesis (IV), and Conclusio (V); structural reorganization only, conditional-reduction status unchanged
• 2.5 (2026-05-24): Public repo synced to the Part-III excluded-paths catalog extension; Part III EN/DE and combined PDFs rebuilt; local agent-control files kept private via .gitignore
• 2.4 (2026-05-23): Parallel programme added to Part III; companion Even-Dominance DOI updated to 10.5281/zenodo.20291994
• 2.3 (2026-05-15): Conditioning clarified from unconditional closure to conditional reduction; public 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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Dieses Projekt ist eine unentgeltliche Open-Source-Schenkung im Sinne der §§ 516 ff. BGB. Die Haftung des Urhebers ist gemäß § 521 BGB auf Vorsatz und grobe Fahrlässigkeit beschränkt. Ergänzend gelten die Haftungsausschlüsse aus GPL-3.0 / MIT / Apache-2.0 §§ 15–16 (je nach gewählter Lizenz).

Nutzung auf eigenes Risiko. Keine Wartungszusage, keine Verfügbarkeitsgarantie, keine Gewähr für Fehlerfreiheit oder Eignung für einen bestimmten Zweck.

This project is an unpaid open-source donation. Liability is limited to intent and gross negligence (§ 521 German Civil Code). Use at your own risk. No warranty, no maintenance guarantee, no fitness-for-purpose assumed.