The Zeta Zoo: The Mathematical Side of Functional Stability Theory

Functional Stability Theory (FST) identifies a single structural pattern — functional positivity under a gauge constraint — as the common substrate of open problems in number theory, mathematical physics, and cosmology. This paper is the mathematical classification side of FST. DRAFT version. This preprint is subject to revisions. Version 2.1 is a guardrail release: it sharpens the status language after strict review, separates self-adjoint YES cases from resonance YES cases, and makes the Riemann, Dedekind, Ihara, CRM, and companion-program claims more explicit. We formalise the trinity of meta-principles governing the zeta-type branch: the Universal Convexity Uniqueness (UCU) lemma, the Semigroup-Group Equivalence (SGE) conjecture, and the branch-transversal Weil quadratic form. Within this trinity, we classify the zeta zoo — a taxonomy of zeta-type families along the SGE dichotomy (HP-BL-YES / HP-BL-NO / HP-BL-OPEN). Instances covered include Riemann zeta, Selberg zeta, Prime-Hub, CRM flow-zeta, Dedekind zeta, Ihara-Petersen graph zeta, and Dirichlet L-function consequences. The paper is a classification and boundary theorem, not a standalone proof of every listed open problem. The Five Masters MasterTitleRoleConcept-DOI ZookeeperThe Spectral ZookeeperCCM microcluster route10.5281/zenodo.19673126 Zeta ZooThe Zeta ZooClassification framework (SGE taxonomy, Boundary Theorem)This paper Spectrum DualityFST Spectrum Duality / RFEPPhysical instantiation (Pattern A, DS1–DS3)10.5281/zenodo.19036190 AtlasDirichlet Character AtlasWeil-kernel Galerkin diagnostics and negative method validation10.5281/zenodo.19960809 SelbergNE-B Failure as Hilbert–Polya DetectionSGE-YES validation (Casimir / Laplace-Beltrami)10.5281/zenodo.19962588 Zoo Population (v2.1) FamilyHP-BL ClassMechanism Riemann zetaNOprime-shiftcondNE-A plus finite NE-B evidence; companion-program closure remains separated Selberg zetaYESsaCasimir / Laplace-Beltrami self-adjoint operator Prime-HubOPENOP5 construction under documented obstructions CRM flow-zetaYESresRuelle-type / Lie group flow with resonance analogue Dedekind Q(sqrt(-5))SGE-predicted NOFinite prime-ideal semigroup evidence Ihara-PetersenYESsaBass-Hashimoto matrix / finite graph certificate Selected FST Foundation Links PaperRoleConcept-DOI FST Hubprogramme umbrella10.5281/zenodo.20130499 RH Even Dominance / RH Trilogyspectral-stability prototype10.5281/zenodo.19035640 FST Spectrum Duality / RFEPphysics master principle10.5281/zenodo.19036190 Dirichlet Character Atlasnegative method validation10.5281/zenodo.19960809 Selbergpositive SGE validation10.5281/zenodo.19962588 Code PDF metadata is populated for EN, DE, and combined files. Version 1.1 (May 2026) Initial public Zenodo release as one of the five FST Master Papers. Live record contained only the English PDF and required later metadata/file-set completion.