RH/GRH Rigidity Program v5.0 Bundle: Main Closure Papers and Auxiliary Audit Notes

## 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; - \(W_2\)-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 \(ζ(s)\)**;- **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.