NE-B Failure as Hilbert-Pólya Detection: Universality of the v2.0 Weil Quadratic Form on the Selberg Zeta Function (DRAFT)

DRAFT version. This is an advanced preprint draft. The paper is published for citation and discussion; further revisions remain possible. Abstract. We test the v2. 0 methodology recently proposed for the Riemann Hypothesis on the Selberg zeta function Sel (s) of a compact hyperbolic surface Γ. The v2. 0 method reduces RH to even dominance of the Weil quadratic form QW_λ via three ingredients: the Shift Parity Lemma, frontier-prime dominance, and two non-existence theorems (NE-A: non-positivity of the prime shift multiplier; NE-B: no universal commuting operator). On the Selberg side: (i) the Shift Parity Lemma transfers unchanged (purely algebraic) ; (ii) frontier dominance transfers with modified constants for the exponential geodesic density eT/T; (iii) NE-B fails for Selberg — the Laplace-Beltrami operator Δ is a universal commuting operator for all geodesic-shift transfer operators, by isometry invariance. This confirms the v2. 0 framework as a precise SGE-classifier: where a classical commuting operator exists (Selberg, via Casimir), v2. 0 is redundant but valid; where it does not (Riemann, by NE-B), v2. 0 is the only available route. The Weil quadratic form is the universal structure; the operator is optional. Hilbert-Pólya is recovered as a special case of v2. 0. Selberg thus serves as the SGE-YES calibration point of the FST programme — the positive control case that complements the Atlas (negative micro-cartography of Dirichlet) as the method-validation pair of the CoreCore. Changes in Version v0. 3 (May 2026) Major: The strict Selberg critical-line statement is now formulated as a spectral zero description plus the explicit no-exceptional-eigenvalue condition λ1 (Δ) ≥ 1/4, rather than as an unconditional theorem. Minor: The Ihara 1966 bibliography entry was corrected to Journal of the Mathematical Society of Japan 18 (1966), no. 3, 219--235. DE/EN: EN/GER/Kombi synchronized; all three PDFs rebuilt and hash-checked. German PDF metadata now uses real Unicode umlauts. 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) 10. 5281/zenodo. 19960809 Selberg NE-B Failure as Hilbert–Pólya Detection SGE-YES validation (v2. 0 universality, Casimir / Laplace-Beltrami) This Paper 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) Dirichlet Character Atlas — Mikro-Kartierung des Zeta-Zoos via Weil-Kernel Galerkin diagnostics (negative method validation) (Concept-DOI: 10. 5281/zenodo. 19960809) This paper — Selberg (NE-B Failure) — SGE-YES validation: v2. 0 universality on Selberg zeta, NE-B fails (positive method validation) 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