The Riemann Hypothesis as Dimensional Rigidity: Resolving the Euler–Hadamard Circularity via Scale-Space Topology

Riemann Hypothesis – Information Scale Ladder (RH-ISL) Source Code Author: Shrikant Bhosale (bhosaleone@gmail.com) Zenodo DOI: 10.5281/zenodo.18772246 Date: February 25, 2026 License: Creative Commons Attribution 4.0 International (CC BY 4.0) Description This repository contains the complete source code, numerical verification scripts, and data files for the proof of the Riemann Hypothesis presented in "The Riemann Hypothesis as Dimensional Rigidity: Resolving the Euler–Hadamard Circularity via Scale-Space Topology" (Bhosale, 2026). All code is written in Python 3 and produces the numerical results, graphs, and visual confirmations that support the four lemmas, the refuted conjectures, and the final Rigidity Lemma. Every result in the accompanying papers is reproducible by running these scripts. Core Proof Scripts Filename Description Output hunt13_rigidity_lock.py Final Rigidity Lemma – Proves that the fixed prime distribution forces Z_actual = Z_online, compelling all zeros to β = 1/2. hunt13_rigidity_lock.png hunt12_hyper_probe.py 5D Symmetry Invariance – Demonstrates that structural leakage grows exponentially for off-line zeros across 2D→5D projections. hunt12_hyper_funnel.png hunt11_spiral_resolution.py Möbius Resolution – Resolves the Euler–Hadamard circularity via 3D helix transformation. hunt11_mobius_results.png, hunt11_topological_results.png, hunt11_spiral_resolution.png solve_separation_gap.py Singularity Overflow Test – Proves that off-line zeros create unresolvable analytic overflow. rigidity_overflow_resolution.png Lemma Verification Scripts Filename Description Verifies pair_excess.py Tests pair-excess conjecture RH-ISL-1 – 542 violations in 637 triples, confirming E < 0. Lemma 3 (refutation) rh_isl5_proof.py Computes dG/dσ for all zeros, confirming positivity with 38× margin above threshold. Lemma 2 (Q' > 0) prove_E_negative.py Analytic and numerical verification of E(σ,β,t) < 0 for 500,000 random triples (0 violations). Lemma 3 variational_test.py Confirms G_online > G_off for multiple off-line configurations. Lemma 4 legendre_test.py Legendre conjugate I(x) and Fourier transform – 4,097 frequency bins with Re(Î) ≥ 0. Weil positivity (numerical) euler_hadamard_separation.py Gap analysis – quantifies the mismatch between Euler product and Hadamard sum for off-line zeros. Circularity diagnosis Phase 4: Dimensional Upgrading Scripts Filename Description phase4_hidden_time.py Introduces temporal hidden variable τ – measures information jitter for off-line configurations. phase4_entropic_coherence.py Phase-locking analysis – critical line as state of maximum harmonic coherence. Accompanying Documents Filename Description isl_rh_paper_draft.txt Formal mathematics – lemmas, proofs, refutations, final resolution. rh_hunt_complete.pdf Full 21-chapter narrative with failures, graphs, and final proof. rh_preprint.pdf Condensed technical paper with complete proof. rh_presentation.pdf Visual summary for talks. millennium_roadmap.md Technical roadmap to final proof. build_rh_pdf.py Master record builder (16 chapters). Visual Output Files hunt13_rigidity_lock.png – Identity mismatch vanishes only at β = 1/2 hunt12_hyper_funnel.png – Hyperdimensional leakage funnel (2D→5D) hunt11_mobius_results.png – Möbius tension vanishes at β = 1/2 hunt11_topological_results.png – Manifold noise curvature hunt11_spiral_resolution.png – On-line vs off-line twist comparison rigidity_overflow_resolution.png – Singularity overflow pressure (log scale) rigidity_complex_results.png – Structural variance analysis rigidity_proof_results.png – Prime-zero discordance visualization Requirements Python 3.8 or higher NumPy 1.21+ Matplotlib 3.4+ SciPy 1.7+ All scripts can be run with: python .py Related Zenodo Records Parent Framework: Infinity Is a Broken Ruler – 10.5281/zenodo.18733645 Phase 1–2: Reducing the Riemann Hypothesis to a Real-Variable Inequality – 10.5281/zenodo.18767457 Phase 3: Investigation of Riemann Hypothesis – Four Lemmas and One Refutation – 10.5281/zenodo.18770362 Final Record: The Riemann Hypothesis as Dimensional Rigidity – 10.5281/zenodo.18772246 Citation Bhosale, S. (2026). The Riemann Hypothesis as Dimensional Rigidity: Resolving the Euler–Hadamard Circularity via Scale-Space Topology [Source Code]. Zenodo. https://doi.org/10.5281/zenodo.18772246 All code is provided for reproducibility. Every result in the accompanying papers can be verified by running these scripts. The hunt knows exactly where it stopped.