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

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 v0.5 (July 2026) Major: The NE-B discussion is now restricted to the spherical Selberg/Casimir convolution channel. Individual geodesic-flow pullbacks are no longer identified with truncated interval shifts, and the erroneous phase-eigenvalue wording has been removed. Major: The conditional v2.0 Selberg-RH theorem now separates the self-adjoint interval realization (C2a), the simple-even minimizer/parity gate (C2b), the factor/window identification ledger (C2c), and the no-exceptional-eigenvalue spectral-gap input (SG/NoExc). Major: New subsection "C2c window ledger with real spectral data": the C2c window certificate is calibrated on real, independently computed spectra (Bolza surface, certified eigenvalues of Strohmaier–Uski 2013; modular surface, rigorous cuspidal Maass data of Seymour-Howell 2022 and the rigorous database of Lowry-Duda 2025) as a finite-spectrum proxy with explicit tail budgets. All positive controls pass after tail adjustment; all negative controls fail, including the mandatory same-Rayleigh wrong-tail class. No claim upgrade: the Connes-minimizer identification remains open and C2c keeps companion/audit status. Major: The zero statement is explicitly scoped to non-trivial spectral zeros in the critical strip, and the open C2c status is treated as a Companion/Audit gate rather than as a completed proof upgrade. Minor: An AI disclosure section (extensive AI-assisted and computational workflow, including model and workflow assignment) has been added to both language versions. Minor: The bibliography maintenance from the local v0.5 candidate is included, in particular the update of the related RH Part II reference to Zenodo v2.4 / DOI 10.5281/zenodo.20358728, and three new web-verified references (Strohmaier–Uski 2013, Seymour-Howell 2022, Lowry-Duda 2025). DE/EN: EN/GER/Kombi synchronized after the 2026-06-23 strict guardrail review and the 2026-07-03 real-data ledger integration; all three PDFs rebuilt and hash-checked, with hard LaTeX log checks clean and German umlauts verified in the visible PDF text. 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 Conditional RH-reduction programme 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 Micro-cartography (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 — conditional RH-reduction programme 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 — micro-cartography of the Zeta Zoo 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 Landscape/Atlas programme (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 conditional-reduction programme. The microcluster closure of CCM step MS2 is a conditional target with open uniform gates, not a proof of RH. 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/tree/main/masters/selberg Other Recommended reading: A Conditional Reduction of the Riemann Hypothesis to Even Dominance of the Weil Quadratic Form — Concept DOI: 10.5281/zenodo.19764771 From Landscape to Atlas: Multi-Route Cartography of an Ongoing Expedition Toward the Riemann Hypothesis — Concept DOI: 10.5281/zenodo.19035640