# OverviewThis record releases a **two-piece program** toward RH(ζ) built around an *unsmoothing bridge* and a**density–0 elimination gate**. The key clarification in v3.9r is the strict separation: - **Circle–A (core):** proves the **density–1** unsmoothing bridge (a rigorous analytic–variational core).- **Density–0 Proof (closure gate):** removes the remaining **density–0 exceptional windows** obstruction at large height via **Gate (A)** and completes a **fixed-target** closure for ζ via **Gate (B)**. The intent is to prevent the most common misread: *Circle–A alone does not claim all-window closure*;the all-window upgrade is imported from the density–0 companion. # Contents of this record- **RH Circle A v3.1r (PDF)**: density–1 core, clarified roadmap levels (core vs refinement vs program-level closure).- **Density–0 Proof v3.9r (PDF)**: Gate (A) + Gate (B) lemmas and fixed-target closure route for ζ.- **RH Guide v1.2r (PDF)**: navigation + one-page dependency table to block misinterpretations.- **Selberg Class Extension v1.0r (PDF)**: auxiliary note (version-aligned to the set). # What is new in v3.9r- **Circle–A roadmap de-contradicted**: explicit 3-level structure: (i) density–1 core, (ii) density–(1−ε) optional refinement, (iii) all-window closure imported via Gate (A) from the companion.- **Gate (A) completion sealed by a dyadic-mesh lemma**: a mechanical “bad window ⇒ bad registry window” transfer lemma is inserted, so that registry control upgrades to all sufficiently large windows without informal leaps.- **Gate (B) completion sealed by remainder→slope→energy lemmas**: remainder vanishing implies slope vanishing, and slope=0 forces the unique minimizer under H2.- **Strong convexity inequality direction corrected** in the slope/energy gap lemma (now consistent with standard λ-convex metric slope bounds).- **One-page dependency table added** to the Guide to prevent conflating Circle–A and program-level closure. # Closed results (as released)- Circle–A provides the **density–1** unsmoothing bridge as the canonical analytic–variational core.- Density–0 Proof provides: - **Gate (A)**: elimination of the density–0 exceptional windows obstruction at large height (via dyadic mesh spike-transfer + EF+LS pointwise control on the registry). - **Gate (B)**: fixed-target closure route for ζ using remainder→slope→energy and strict convexity. # Dependency table (anti-misread, summarized)- Density–1 unsmoothing bridge: proved in Circle–A (core).- All-window upgrade at large height: proved in Density–0 Proof (Gate A), imported into the program.- Fixed-target ζ closure: proved in Density–0 Proof (Gate B + main closure theorem).- Finite-height verification: optional and logically separate. # Notes- The all-window claim is a **program-level** statement: Circle–A does not assert it in isolation.- Version alignment in this record: Circle–A v3.1r, Density–0 Proof v3.9r, Guide v1.2r, Selberg extension v1.0r. ***Source availability policy: This is a PDF-only public release at the current stage.Full LaTeX sources, patch bundles, and reproducible build notes are maintained as a controlled companion artifactand can be provided to journal referees or independent verifiers upon request. ==================================== Author :Byoungwoo Lee (leeclinic@protonmail.com / Daejeon, South Korea)
Byoungwoo Lee (Wed,) studied this question.