The Dirichlet Character Atlas — A Weil-Kernel Numerical Atlas for Dirichlet L-Function Sector Gaps (Galerkin Diagnostics and the Boundaries of Leading-Order Theory) (DRAFT)

DRAFT version. This is an advanced preprint draft. The paper is published for citation and discussion; further revisions remain possible. Abstract. This paper develops a Weil-kernel numerical atlas of sector gaps for ten real Dirichlet characters (D ∈ 5, 8, 12, 13, 17, 21, 24, 29, 33, 60) at Galerkin resolutions N ∈ 200, 400, 600. Main results are negative but structurally informative: the leading-order approximation φ+ ≈ φ- is empirically falsified, the full formula with Galerkin ground states is a matrix-decomposition tautology, and χ21 oscillates as an N-truncation artifact. The paper positions the Weil-kernel architecture as a precise diagnostic tool and roadmap for the Dirichlet-twisted CCM Route D (Zookeeper follow-up). Within the FST programme, the Atlas serves as the micro-cartography companion to the macro-classification paper (Zeta Zoo): both are CoreCore Master papers and complement each other. The Five Masters Master Title Role DOI (Concept) Zookeeper The Spectral Zookeeper RH proof via CCM microcluster closure 10. 5281/zenodo. 19673126 Zeta Zoo The Zeta Zoo Classification (SGE taxonomy, Boundary Theorem) 10. 5281/zenodo. 19673226 Spectrum Duality FST Spectrum Duality / RFEP Physical instantiation (Pattern A, DS1–DS3) 10. 5281/zenodo. 19036190 Atlas Dirichlet Character Atlas Mikro-Kartierung (Galerkin diagnostics; negative method validation) This Paper Selberg NE-B Failure as Hilbert–Pólya Detection SGE-YES validation (v2. 0 universality, Casimir / Laplace-Beltrami) 10. 5281/zenodo. 19962588 Series information One of five FST Master Papers (functional positivity, spectral, classification, atlas, validation): Zookeeper — RH proof via spectral microcluster closure (CCM Fourier model) (Concept-DOI: 10. 5281/zenodo. 19673126) The Zeta Zoo — Mathematical classification via SGE taxonomy (Concept-DOI: 10. 5281/zenodo. 19673226) FST Spectrum Duality / RFEP — Physical instantiation (Pattern A, DS1-DS3) (Concept-DOI: 10. 5281/zenodo. 19036190) This paper — Dirichlet Character Atlas — Mikro-Kartierung des Zeta-Zoos via Weil-Kernel Galerkin diagnostics (negative method validation) Selberg (NE-B Failure) — SGE-YES validation: v2. 0 universality on Selberg zeta, NE-B fails (positive method validation) (Concept-DOI: 10. 5281/zenodo. 19962588) Glossary — FST core terms TermMeaning v2. 0 Method package developed in the RH programme (RH Trilogy v2. 1, Concept-DOI 10. 5281/zenodo. 19035640): reduces RH to even dominance of the Weil quadratic form QWλ via four ingredients — the Shift Parity Lemma, frontier-prime dominance, and the two non-existence theorems NE-A and NE-B. NE-A Non-existence theorem A. The Fourier multiplier of the prime shift operator Aλ on the critical line is non-positive — the multiplier cannot serve as a positive-definite (Hilbert–Pólya) operator. NE-B Non-existence theorem B. No universal symmetric operator commutes with all Shift-Parity difference matrices DN (r) ; the only common commutant is a scalar multiple of identity (computer-assisted proof for N ≤ 15). Together with NE-A this rules out the classical Hilbert–Pólya route — and is exactly why v2. 0 is needed for Riemann. SGE Semigroup–Group Equivalence. Classification axis of the Zeta Zoo: HP-BL-YES (a classical commuting operator exists, e. g. Casimir for Selberg), HP-BL-NO (commutant blocked, Riemann case), HP-BL-OPEN (undecided, e. g. Prime-Hub). Weil quadratic form QWλ Truncated explicit-formula quadratic form whose positivity controls the location of zeros. Universal across the zeta zoo; the operator behind it is family-dependent (and may be absent — see NE-B). Hilbert–Pólya Conjecture that the Riemann zeros are eigenvalues of a self-adjoint operator. v2. 0 generalises this: where Hilbert–Pólya works (SGE-YES, e. g. Selberg via Casimir), v2. 0 reproduces it; where it fails (SGE-NO / NE-B, Riemann case), v2. 0 still applies. Pattern A Functional Positivity under a Gauge Constraint — the universal stability pattern of FST. Instantiated in physics (Yang-Mills mass gap, Navier-Stokes), cosmology (Dark Energy / Hu-Sawicki), and via SGE in the zeta-type branch. RFEP Renormalized Free-Energy Principle. Mathematical core principle of FST; supplies the dissipative selection axioms DS1–DS3. CCM Connes–Consani–Moscovici. Fourier model for the Weil quadratic form used in the Zookeeper proof. The microcluster closure of CCM step MS2 is the technical core of the unconditional RH proof. UCU Universal Convexity Uniqueness lemma. Together with SGE and the Weil quadratic form, the trinity of meta-principles governing the zeta-type branch (Zeta Zoo). Technical info Repository: https: //github. com/research-line/functional-stability-theory Other Recommended reading: The Riemann Hypothesis: A Direct Proof via Even Dominance of the Weil Quadratic Form — Concept-DOI: 10. 5281/zenodo. 19764771 RH Even Dominance v2. 1 (Trilogy, Part I-III) — Concept-DOI: 10. 5281/zenodo. 19035640 Changes in Version v2 (May 2026) Source correction: The LMFDB passage was narrowed from a broad zero-certificate claim to computed low-lying zeros as finite consistency data. Metadata: EN/DE/Kombi PDFs were rebuilt and PDF metadata was normalized before upload. Files: The public upload contains refreshed EN, DE and combined PDFs.