Invetigation of Riemann Hypothesis

This repository contains the complete technical record, research scripts, and analytic proofs for the RH-ISL project (Phase 3). The project introduces an information-theoretic framework for the Riemann zeta function and derives four rigorous lemmas that constrain the distribution of non-trivial zeros. While the project does not prove the Riemann Hypothesis, it successfully reduces it to a precisely isolated gap (the Prime–Zero Separation Problem) and provides the first information-geometric derivation of variational bounds on Riemann zeros. 2. Contents /docs rh_hunt_complete.pdf — The 15-chapter Feynman-style comprehensive project record. Includes all proofs, graphs, methods, numerical tests, refutations, and documented failure modes. isl_rh_paper_draft.txt — A publication-ready summary of the four proved lemmas and the precise diagnosis of the remaining gap. /scripts (Reproducibility Core) lemma3_complete_proof.py — Rigorous analytic proof of Pair-Excess Negativity (E < 0). rh_isl5_proof.py — Proof that Q'(σ) > 0 and Q(σ) < 0 on the critical strip. variational_test.py — Large-scale numerical verification (500,000 triples) of the variational bound. euler_hadamard_separation.py — The “Separation Attack” script identifying the circularity gap. gue_spacing.py — Analysis of Riemann zero spacings confirming GUE level repulsion. pair_excess.py — Numerical refutation of the initial RH-ISL-1 conjecture. /build build_rh_pdf.py — Python script used to generate the final project PDF. Requires reportlab, matplotlib, and numpy. /data_archive Technical notes, intermediate drafts, and abstracts from earlier phases of the hunt. 3. How to Reproduce Each script in the /scripts directory is standalone. To verify the results: 1. Analytic Lemma 3 Verification python scripts/lemma3_complete_proof.py 2. Variational Numerical Test python scripts/variational_test.py 3. GUE Spacing Analysis python scripts/gue_spacing.py To rebuild the comprehensive PDF pip install reportlab matplotlib numpy cd build python build_rh_pdf.py 4. The Four Proved Lemmas Lemma 1: H''(σ) = 4 Var[log n] > 0 (Entropy Convexity). Lemma 2: Q(σ) < 0 for all σ ∈ (1/2, 1) (RH-ISL-5). Lemma 3: E(σ, β, t) < 0 for all off-line zero pairs (Pair-Excess Negativity). Lemma 4: On-line zero configurations maximize the information-balance G-sum. 5. Status & The Missing Gap The project concludes that the Riemann Hypothesis is uniquely tied to the analytic rigidity of the Euler product continuation. The circularity of assuming Qactual = Qonline is explicitly documented. The remaining open problem is reduced to proving that the prime distribution forces Q(σ) into its G-maximizing state. Status: Provisionally ClosedAuthor: Shrikant Bhosale (bhosaleone@gmail.com)Zenodo Reference (Phase 1): https://zenodo.org/records/18767457