Riemann Hypothesis Program: Circle A Construction and Density–0 Elimination (v3.9 set)

# OverviewThis record releases the **v3.9 set** of a variational/rigidity program toward the Riemann Hypothesis (RH) and related Selberg-class targets. The set is organized as a **Circle A construction** (rigorous in the smoothed and density–1 regimes) plus a companion **Density–0 elimination** component that upgrades the program to an exception–free large-height statement via explicit “Gate (A)/(B)” lemmas. **Core principle.** A rigidity functional \({G}(F)\) is constructed so that a closure law of the form\[{G}(F)=0 RH(F)\]can be enforced through an Euler–Lagrange remainder controlled by EF+LS upgrades and EVI-type gradient-flow rigidity. # What is new in v3.9- **Gate (A)/(B) made explicit and proved (Appendix C.2A of Density–0 Proof):** - Defect observables \(D_F(T)\) and \(E_F(T)\) are defined. - A **persistence lemma** upgrades a pointwise violation into an \(L^2\) lower bound using band-limited Lipschitz control. - A **strong-convexity gap lemma** quantifies the slope–energy relation under uniform \(λ_0\)-convexity/EVI. - **Gate (A)**: density–1 control \(⇒\) exception–free control for all sufficiently large windows. - **Gate (B)**: once Gate (A) holds for the trivial degree–1 member \(F=ζ\), the closure law yields \(RH(ζ)\).- **Appendix C.3 meta-theorem restored:** the previously referenced density–0 elimination theorem is now included as a concrete theorem/proof block.- **Set hygiene:** file-name/version consistency is aligned across Circle A / Density–0 / Guide / Selberg note. # Scope & non-toy status- This record is a **theory-first** release: PDF sources are included for full auditability.- The role split is explicit: Circle A establishes the smoothed/density–1 regime; Density–0 Proof discharges the density–0 exceptional-window obstruction at large height through Gates (A)/(B). # How to citeCite this Zenodo record as the v3.9 set, and cite component PDFs by their filenames and version tags in the manuscript text. # Artifact integrity (PDF checksums) SHA256 (PDF-only upload):- Density-0_Proof_v3.9.pdf: 8536d2fbab2fffbd78b16d6d40bebce7822ad8d068640c47e4243baf021b1ba3- RH_Circle_A_v3.1.pdf: d19d5198143c8f267fa19e90b6470062186a1d2c0b338c14504e12590e465c40- RH_Guide_v1.2.pdf: 3bb2a8bb05a00cf7ea93a89a463e813f567b915eab3b40daf16c9061610a736a- Selberg_Class_Extension_v1.0r.pdf: ebba3f065498ad365ee79bc9dc3737fca54c74b17415412f653e230f5f95ab2b Note: This Zenodo version is a **PDF-only release** for reader convenience.TeX sources and patch bundles are maintained separately and can be provided as a companion artifact if needed. ==================================== Author Byoungwoo Lee (leeclinic@protonmail.com)