## Overview This record is the **curated bundle release** of the RH/GRH rigidity program at the v5. 0 stage. Unlike the main theorem-bearing records, this bundle is not a single paper. It is a **reader-facing audit packet** that gathers the main v5. 0 closure papers together with the auxiliary standalone notes extracted from the closure chain. Its purpose is to make the theorem-level architecture of the verified degree-\ (2\) package easier to read, navigate, and audit as a single packet. The bundle should be read as a **modular closure set**, not as a new monolithic paper. ## What is included in this bundle The bundle contains two layers. ### A. Main v5. 0 closure paper set (4 documents) These are the main theorem-bearing papers of the v5. 0 release: - **RH Circle A v5. 0** rigorous smoothed / density-1 core, together with the distortion-stable \ (R\) -circle reduction; - **Density-0 Proof v5. 0** the two closure gates not claimed inside Circle A itself: \ Gate (A): \ density-1 / log-density-1 control all sufficiently large windows, \ \ Gate (B): \ remainder metric slope rigidity energy each fixed admissible target; \ - **RH Guide v5. 0** reader-facing dependency tables, jump ledgers, and minimal audit paths; - **Selberg Class Extension v5. 0** an optional extension layer, separate from the verified degree-\ (2\) closure spine. ### B. Auxiliary standalone audit notes (5 documents) These notes extract the most load-bearing reviewer-facing jumps into narrower audit objects: - **Closure Set Overview v1. 1** all versions DOI: **10. 5281/zenodo. 18934096** - **Gate (A) standalone refined note v1. 3** all versions DOI: **10. 5281/zenodo. 18922927** - **H3-A: Arithmetic Pointwise Upgrade v1. 4** all versions DOI: **10. 5281/zenodo. 18934195** - **H3-B v1. 6A Companion Set v1. 6A** (Rigidity Elimination for H3 v1. 6) + (Burden A: Superlevel-Closedness v1. 1) all versions DOI: **10. 5281/zenodo. 18934229** - **Gate (B): Fixed-Target Variational Closure v1. 7** all versions DOI: **10. 5281/zenodo. 18934285** These auxiliary notes do not replace the main v5. 0 closure papers. Their role is to reduce reviewer burden by isolating specific load-bearing jumps from the full package. ## Verified low-degree scope Within the declared admissible parameter box, the verified scope of the v5. 0 release is degree \ (2\): - the Riemann zeta function;- primitive Dirichlet \ (L\) -functions;- holomorphic \ (GL (2) \) newforms. In this regime, the closure is organized into three theorem layers: 1. **H1/H2/H3 package** - well-posedness of the renormalized rigidity functional; - \ (W₂\) -geodesic displacement convexity; - explicit-formula plus large-sieve cancellation, together with the arithmetic pointwise upgrade package; 2. **Gate (A): all-window upgrade** - elimination of density-0 exceptional windows at large height; 3. **Gate (B): fixed-target closure** - remainder \ (\) slope \ (\) energy vanishing for each admissible target. According to the theorem-level closure package carried by the v5. 0 main papers, this yields: - **RH for \ ( (s) \) **;- **GRH for primitive Dirichlet \ (L\) -functions and holomorphic \ (GL (2) \) newforms**. Equivalently, the v5. 0 stage presents a theorem-level rigidity closure package in verified degree \ (2\), together with a modular audit decomposition of its most load-bearing closure steps. ## Why this bundle exists The v5. 0 architecture is intentionally modular. The main papers already separate: - the density-1 / smoothed Circle A core, - the density-0 elimination step, - the fixed-target variational closure step, - and the optional higher-degree extension agenda. The auxiliary notes in this bundle push that modularity one level further by turning several reviewer-facing burdens into standalone audit notes. In particular, the bundle separates the closure path into: - an **arithmetic half** of H3 (mean-to-pointwise upgrade), - a **rigidity-elimination half** of H3 (no-exception seal), - a **fixed-target variational half** (Gate (B) ), - and an **auxiliary upstream Gate (A) note**. This makes the reading and checking order significantly lighter than treating the whole closure path as a single undifferentiated manuscript. ## Suggested reading order A referee or reader may use the following order. ### Minimal entry order1. **Closure Set Overview v1. 1**2. **RH Guide v5. 0**3. **Density-0 Proof v5. 0** ### Auxiliary-note order1. **H3-A: Arithmetic Pointwise Upgrade**2. **H3-B: Rigidity Elimination for H3**3. **Gate (B): Fixed-Target Variational Closure**4. **Gate (A) standalone refined note** (upstream auxiliary note) ### Main-package order1. **RH Circle A v5. 0**2. **Density-0 Proof v5. 0**3. **RH Guide v5. 0**4. **Selberg Class Extension v5. 0** (optional) ## What is optional and what is not The verified closure spine in degree \ (2\) is carried by the main v5. 0 theorem package. The following are **not** part of the verified low-degree spine: - the higher-degree Selberg-class extension agenda;- optional interface or proof-skeleton appendices;- finite-height verification or numerical validation modules. These remain modular and separate from the theorem-level verified degree-\ (2\) closure. ## Higher-degree Selberg extension Any extension beyond degree \ (2\) is intentionally separated from the proof spine of the present bundle. For higher-degree Selberg-class families, the remaining gap is reduced to explicit family-uniform inputs, including: 1. uniform Archimedean curvature lower bounds for the relevant gamma-factor geometry;2. family-uniform large-sieve constants at the explicit-formula band limit. These higher-degree requirements are isolated as an **optional extension layer** and do **not** enter the verified degree-\ (2\) closure carried by the main v5. 0 set. ## DOI guidance This record is the **bundle snapshot** for the curated packet. - **Latest curated bundle entry (concept DOI): ** **10. 5281/zenodo. 17075435** - **Current curated bundle snapshot (version DOI): ** **10. 5281/zenodo. 18933975** Bundle releases are identified by the suffix `bundle` in the version string/title. Use the **version DOI** when citing this exact curated bundle snapshot. Use the **concept DOI** when directing readers to the latest curated bundle entry. The auxiliary notes listed above remain independently citable by their own all-versions DOIs. ## Summary This record should be read as the **curated v5. 0 bundle** of the RH/GRH rigidity program: a theorem-level verified degree-\ (2\) closure package together with a set of narrowly scoped standalone audit notes that isolate the main reviewer-facing jumps. It is not a new monolithic proof paper. It is a structured audit bundle designed to make the v5. 0 closure architecture easier to access, cite, and review. ==================================== Author: Byoungwoo Lee (leeclinic@protonmail. com / Daejeon, South Korea)
Byoungwoo Lee (Tue,) studied this question.
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