The Zeta Zoo: Branch-Local Hilbert-Polya via the Trinity of UCU, SGE, and Weil-Form Transversality

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. 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: Riemann zeta (NE-A, NE-B obstructions), Selberg zeta (Casimir operator), Prime-Hub (three obstructions), CRM flow-zeta, Dedekind zeta, Ihara-Petersen graph zeta. Eight further families are discussed in the outlook. The Three Masters Master Title Role DOI Zookeeper The Spectral Zookeeper RH proof via CCM microcluster closure 10.5281/zenodo.19673126 Zeta Zoo The Zeta Zoo Classification (SGE taxonomy, Boundary Theorem) This Paper Spectrum Duality FST Spectrum Duality / RFEP Physical instantiation (Pattern A, DS1–DS3) 10.5281/zenodo.19162705 Zoo Population (v1.1) Family HP-BL Class Mechanism Riemann zeta NO (classical) NE-A + NE-B obstructions Selberg zeta YES Casimir / Laplace-Beltrami Prime-Hub OPEN Three obstructions documented CRM flow-zeta YES Ruelle-type, Lie group flow Dedekind Q(sqrt(-5)) NO Prime-ideal semigroup Ihara-Petersen YES Bass-Hashimoto matrix Domain Supplements — FST-Mathematics Classified by the SGE taxonomy from the Zeta Zoo. BSD, Hodge, and P vs NP are bridge species (math + physics). Paper Version Status Open Problem DOI BSD v1.1 Reformulation Higher Gross–Zagier (rank ≥ 2) 10.5281/zenodo.19087443 Hodge v1.1 No-Go Theorem = Deligne’s question (1982) 10.5281/zenodo.19087439 P vs NP v1.2 Reformulation Uniformity Bridge 10.5281/zenodo.19056809 Domain Supplements — FST-Physics Derive Pattern A + DS1–DS3 from Spectrum Duality. Paper Version Status Open Problem DOI Turbulence v1.3 Journal-ready DFC1 empirical (only input) 10.5281/zenodo.19056813 Yang–Mills v2.1 Conditional Analytical proof of λ < 0 10.5281/zenodo.19087433 Navier–Stokes v2.1 Conditional Assumption G2 (projection) 10.5281/zenodo.19087449 NS Log-Distance v1.3 Proof of Life TLL for 3D NS open 10.5281/zenodo.19056807 Domain Supplements — FST-Cosmology The Dark Energy paper instantiates Pattern A on cosmological screening mechanisms (Hu–Sawicki f(R) gravity). Paper Version Status Open Problem DOI Dark Energy v1.6 Framework Note Hu–Sawicki parameters open 10.5281/zenodo.19036235 FST-Biology: In development. FST-Chemistry: Planned. Proof Architecture THREE MASTERS +---------------------+---------------------+ | | | Zookeeper Zeta Zoo Spectrum Duality (RH proof) (Classification) (Pattern A, DS1-DS3) | | | | FST-Mathematics FST-Physics | | | | +-------+-------+ +-----+-----+ | | | | | | | | BSD+ Hodge+ PvNP+ TU YM NS | | | FST-Cosmology| | | NS-LDI | DE | Status: PROVEN (unconditional, CCM route) + = bridge species (math + physics) Hierarchy FST (Functional Stability Theory) | +-- Masters | +-- Zookeeper RH proof (CCM microcluster closure) | +-- Zeta Zoo Mathematical classification (SGE taxonomy) | +-- Spectrum Duality Physical instantiation (RFEP, Pattern A) | +-- FST-Mathematics BSD, Hodge, P vs NP +-- FST-Physics Turbulence, Yang-Mills, Navier-Stokes, NS-LDI +-- FST-Cosmology Dark Energy +-- FST-Biology (in development) +-- FST-Chemistry (planned) Chronological Development 2025/2026 CRM I-IV (dark energy) RH "light" proof (even dominance) developed independently developed independently \ / +---------+---------+ | Recognition: both share the same structural pattern (Pattern A) | v RFEP formulated (general principle) | Several dead ends | v Idea: classify zeta-type families using techniques from the RH proof | Not enough -- need deeper tools | v RH via Connes framework (CCM) microcluster closure -> unconditional proof | v Zeta Zoo opens: SGE taxonomy classifies all zeta families | v Reorganization under FST (Three Masters + five domain branches) Core Concepts Level Name Acronym Meaning Programme Functional Stability Theory FST The programme name (umbrella) Principle Renormalized Free-Energy Principle RFEP The mathematical core principle Pattern Pattern A: Functional Positivity under Gauge Constraint Pattern A The universal stability pattern