What question did this study set out to answer?

This research aims to validate the v2.0 framework for the Riemann Hypothesis using the Selberg zeta function.

July 5, 2026Open Access

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

Key Points

This research aims to validate the v2.0 framework for the Riemann Hypothesis using the Selberg zeta function.
Test v2.0 methodology on the Selberg zeta function Sel(s).
Analyze the implications of NE-B's failure in relation to the Laplace-Beltrami operator.
Examine changes in the v0.5 and v0.3 versions regarding spectral zeros and geodesic shifts.
NE-B confirms that the Laplace-Beltrami operator is a universal commuting operator for Selberg.
v2.0 remains valid where other methods fail, particularly in Riemann's case.
All positive controls pass after tail adjustment; negative controls do not.

Abstract

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. 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

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