What question did this study set out to answer?

The goal is to establish a variational/rigidity approach that enhances understanding of the Riemann Hypothesis and related problems.

February 20, 2026Open Access

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

Key Points

The goal is to establish a variational/rigidity approach that enhances understanding of the Riemann Hypothesis and related problems.
Introduced Circle A construction for density-1 regimes.
Developed Density-0 elimination using Gates (A)/(B) for large-height statements.
Defined defect observables and persistence lemmas.
Restored a previously referenced density-0 elimination theorem.
The new constructions provide exception-free controls for large windows under density-1 regimes.
The closure law for the trivial case F=ζ reinforces the Riemann Hypothesis.
A stronger quantitative relationship was established between slope and energy.

Abstract

# 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 (F) =0 RH (F) 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 \ (DF (T) \) and \ (EF (T) \) are defined. - A **persistence lemma** upgrades a pointwise violation into an \ (L²\) lower bound using band-limited Lipschitz control. - A **strong-convexity gap lemma** quantifies the slope–energy relation under uniform \ (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=\), the closure law yields \ (RH () \). - **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 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-0Proofᵥ3. 9. pdf: 8536d2fbab2fffbd78b16d6d40bebce7822ad8d068640c47e4243baf021b1ba3- RHCircleAᵥ3. 1. pdf: d19d5198143c8f267fa19e90b6470062186a1d2c0b338c14504e12590e465c40- RHGuideᵥ1. 2. pdf: 3bb2a8bb05a00cf7ea93a89a463e813f567b915eab3b40daf16c9061610a736a- SelbergClassExtensionᵥ1. 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)

Read Full Paperexternally

Mark Helpful

Bookmark

Relay

View Full Paper