Rigidity Program for RH/GRH in Degree ≤ 2: Density–0 Elimination, Circle A, and Gate(A/B) Closure Set, Version v5.0e

This record contains Version v5.0e of the author’s modular closure package for the RH/GRH rigidity program in the verified degree ≤ 2 scope. The bundle is organized around a rigidity framework for eliminating density–0 exceptional windows in the Riemann Hypothesis / Generalized Riemann Hypothesis setting. The central proof carrier is `Density0_Proof_v5.0e`, which develops a renormalized rigidity functional \(G_e\), combines well-posedness, Wasserstein displacement convexity, and explicit-formula/large-sieve cancellation, and records the Gate(A)/Gate(B) closure architecture for the low-degree setting. The package covers the author’s declared degree ≤ 2 scope: the Riemann zeta function, primitive Dirichlet \(L\)-functions, and holomorphic \(GL(2)\) newforms within the stated admissible parameter box. Higher-degree Selberg-class extensions are separated as an optional research layer and do not enter the low-degree closure spine. ### Files in this bundle 1. `Closure_Overview_v1.3.pdf` A reviewer-facing overview and dependency map for the current closure set. It explains the reading order, import conventions, and the final target-facing interface. 2. `Density0_Proof_v5.0e.pdf` Main proof carrier for the density–0 elimination and low-degree closure package. It organizes the proof around H1/H2/H3, Gate(A), Gate(B), and the fixed-target target-detection/front-seal interface. 3. `RH_Circle_A_v5.1.pdf` Circle A / Track A core note. It supplies the smoothed and density–1 unsmoothing bridge, together with the R-packet target-detection mechanism. The exception-free all-window closure is not claimed inside Circle A itself, but imported from the companion Gate(A)/Gate(B) closure package. 4. `H3A_Arithmetic_Upgrade_v1.5.pdf` Arithmetic pointwise-upgrade note. It isolates the passage from imported window-integrated averaged EF+LS cancellation to density–0 pointwise control using local averaging, grid-point mean-square recovery, Chebyshev, and admissible-grid counting. 5. `H3B_Rigidity_Elimination_v1.7.pdf` Rigidity-elimination note for H3. It isolates the no-exception seal from imported density–0 pointwise control, including the Burden B interface certificate between the arithmetic remainder layer and the abstract rigidity-defect layer. 6. `GateA_v1.4.pdf` Reviewer-facing dyadic-registry note for Gate(A). It records the upgrade from density–1 windowwise control to all sufficiently large windows via spike persistence, common-point mesh transfer, and registry-level \(L^2\)-tail budget counting. 7. `GateB_v1.8.pdf` Fixed-target variational closure note for Gate(B). It records the chain \[ D_F(T)→0 |∂ G_e|(μF,T)→0 G_e(μF,T)→0, \] and then formulates the target-facing closure transfer using off-critical detection and the finite-front seal. 8. `RH_Guide_v5.0c.pdf` Reader-accessibility guide and proof-navigation document. It gives the quick-start audit path, dependency ledger, glossary, and background explanations for number theorists, analysts, and graduate readers. ### Main structural changes in v5.0e - H3-A now includes a local-averaging recovery step before applying grid-point Chebyshev estimates.- H3-B now explicitly separates the Burden B compatibility certificate: arithmetic vanishing, positive abstract defect from residual failure, and no detached density–0 cluster component.- Gate(A) is isolated as a standalone dyadic-registry audit note.- Gate(B) is strengthened into a fixed-target variational closure note with an explicit target-detection/front-seal interface.- RH Circle A is reformulated from a fragile R-circle triple criterion into an R-packet target-detection framework.- The overview and guide have been synchronized to the current v5.0e/v5.1/v1.8/v1.7/v1.5/v1.4 structure. ### Scope note The uploaded materials present the author’s theorem-facing closure package and proof architecture. The low-degree RH/GRH claim is stated within the author’s declared verified degree ≤ 2 framework. The higher-degree Selberg-class extension is explicitly treated as optional and conditional on additional family-uniform inputs. ==================================== Author :Byoungwoo Lee (leeclinic@protonmail.