# 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) - `RHCircleAᵥ4. 0. pdf`- `Density-0Proofᵥ4. 0. pdf`- `RHGuideᵥ1. 3. pdf` (compiled for reader accessibility; RH set v4. 0) - `SelbergClassExtensionᵥ1. 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\ ( (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 `RHGuideᵥ1. 3. pdf` (10–15 minutes): roadmap + dependency/jump ledger. 2) Read `RHCircleAᵥ4. 0. pdf` Abstract + the import convention paragraph (to fix the quantifier scope). 3) Read `Density-0Proofᵥ4. 0. pdf` Gate (A) and Gate (B) statements and their sealing substeps. 4) Only if desired: `SelbergClassExtensionᵥ1. 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. com / Daejeon, South Korea)
Byoungwoo Lee (Wed,) studied this question.