RH Rigidity Program Set v4.0 (PDF-only): Circle A Core, Density–0 Gates (A,B), and Guide

# OverviewThis record releases the **public PDF-only** version of the *RH Rigidity Program* set (v4.0).The set is organized as a modular closure pipeline: - **Circle A (core analytic layer / Track A):** rigorous smoothed + density–1 unsmoothing backbone.- **Density–0 Proof (program closure layer):** Gate (A) removes density–0 exceptional windows at large height, and Gate (B) seals the analytic chain “remainder → slope → energy” for fixed-target closure.- **Guide (reader accessibility):** a compact entry point plus a 1-page dependency/jump ledger for audit-style reading.- **Selberg extension (optional):** an additional, explicitly non-essential layer beyond degree ≤2. > **Release policy (important):** This Zenodo record uploads **PDF files only**.> The full LaTeX-source release (the “full set”) is intentionally postponed until further strengthening / journal-ready consolidation. # Contents of this record (PDF-only)- `RH_Circle_A_v4.0.pdf`- `Density-0_Proof_v4.0.pdf`- `RH_Guide_v1.3.pdf` (compiled for reader accessibility; RH set v4.0)- `Selberg_Class_Extension_v1.1.pdf` (optional extension; not used in degree-1 closure) # Program snapshot (how the pieces fit)## Circle A (core)Circle A establishes the unconditional analytic backbone in the smoothed / density–1 regimesvia a Wasserstein-gradient framework (entropy decay, Fisher contraction, and an analytic routeto the R–circle criterion). **Import convention (neutralizing the “all windows” pitfall):**Whenever Circle A states “for all sufficiently large windows,” this is a *program-level* statementimported from the companion Density–0 Proof via Gate (A).Fixed-target closure statements invoked by Circle A are imported via Gate (B).Circle A alone does **not** claim an unconditional all-window upgrade by itself. ## Density–0 Proof (closure gates)- **Gate (A)** (dyadic-spike sealing): upgrades from “density–1 windows” to “all sufficiently large windows”.- **Gate (B)** (remainder → slope → energy sealing): converts vanishing remainder defect into the vanishing rigidity energy required for fixed-target closure. The Gate packaging is designed to be “acceptance-test” friendly (A1–A3 and B1–B2 style subchecks). # Scope & modularityPrimary verified target: **degree ≤ 2 GRH**, covering\(ζ(s)\), Dirichlet \(L\)-functions, and holomorphic \(GL(2)\) newformswithin the declared admissible parameter box.Higher-degree Selberg-class coverage is treated as an **optional** extension layer and is not requiredfor the low-degree closure. # How to read (suggested order)1) Start with `RH_Guide_v1.3.pdf` (10–15 minutes): roadmap + dependency/jump ledger.2) Read `RH_Circle_A_v4.0.pdf` Abstract + the import convention paragraph (to fix the quantifier scope).3) Read `Density-0_Proof_v4.0.pdf` Gate (A) and Gate (B) statements and their sealing substeps.4) Only if desired: `Selberg_Class_Extension_v1.1.pdf` (explicitly optional / conditional layer). # Relationship to the previous public recordThis v4.0 PDF-only record is a continuation of the public Zenodo line (previous public PDF-only: v3.9, 10.5281/zenodo.18677439),with the emphasis in v4.0 on: (i) a clean public-facing release, (ii) Gate(A) dyadic-spike sealing,(iii) Gate(B) remainder→slope→energy sealing, and (iv) a reader/auditor-facing dependency ledger. # Notes- This is a research preprint release (no peer review). Critical scrutiny and independent verification are encouraged.- The LaTeX-source “full set” will be released later after additional consolidation and strengthening. ==================================== Author :Byoungwoo Lee (leeclinic@protonmail.