What question did this study set out to answer?

This paper aims to classify zeta-type families using Functional Stability Theory, focusing on structural patterns among mathematical problems.

July 5, 2026Open Access

The Zeta Zoo: The Mathematical Side of Functional Stability Theory

Key Points

This paper aims to classify zeta-type families using Functional Stability Theory, focusing on structural patterns among mathematical problems.
Formalizes a framework based on three meta-principles: Universal Convexity Uniqueness lemma, Semigroup-Group Equivalence conjecture, and branch-transversal quadratic forms.
Classifies multiple zeta functions including Riemann zeta, Selberg zeta, and others according to the SGE taxonomy.
Introduces a new taxonomy for zeta-type families, highlighting distinctions among YES, NO, and OPEN cases.
Clarifies the boundaries of classification relevant to Riemann and other zeta functions, improving understanding of connections to open problems in number theory.

Abstract

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. Changes in Version 2.3 (July 2026) Maintenance release after the post-v2.2 strict-review hardening and visible version-marker cleanup. This remains a conditional boundary-classification preprint and does not claim a standalone proof of RH or of every listed companion problem. Claim boundaries: SGE is framed as a conjectural audit heuristic / waterline rather than theorem-level closure. Riemann/NE-A guardrail: NE-A compatibility is limited to ordinary positive trace-formula / prime-shift candidates. Boundary taxonomy: adds the Waisen/Ihara two-collar and Sturmian/finite-pattern gate; long finite residue stability remains diagnostic until phase, Sturmian/irrational-rotation, homotopy-winding or true limit arguments are supplied. Version hygiene: EN and DE title pages now carry explicit v2.3 maintenance markers. DE/EN: English, German and combined PDFs are rebuilt synchronously from the v2 source set. Changes in Version 2.2 (June 2026) Maintenance release after the post-v2.1 proof/paper-math and strict-review guardrail passes. This version remains a conditional boundary-classification preprint and does not claim a standalone proof of RH or of every listed companion problem. Claim boundaries: the Riemann three-component passage is now presented as a route ledger RHRoute(X), not as a theorem-level equivalence. Selberg guardrail: HP2 is restricted to the canonical trace-formula / wave-kernel algebra, with group provenance separated from closed geodesics as trace-formula data. Boundary taxonomy: new transfer guardrails distinguish prunable appendages, Schur-cofactor obstruction, fixed-target versus global-transfer obstruction, and local-pass/global-section obstruction. Bibliography and disclosure: the post-v2.1 citation cleanup and neutral L5+ AI disclosure remain incorporated in the synchronized source set. DE/EN: English, German and combined PDFs are rebuilt from the synchronized local v2 source set. Version 2.1 (May 2026) Guardrail review: the classification survived a strict contradiction pass, but the status language was narrowed to distinguish conditional prime-shift NO, self-adjoint YES, resonance YES, and finite-evidence predictions. Claim boundaries: Riemann and Direct Frontier statements are framed as companion-program claims rather than as standalone consequences of this classification paper. Taxonomy sharpened: Selberg, CRM, Dedekind and Ihara are separated into distinct mechanism classes instead of being collapsed into a single YES/NO table. Bilingual sync: English, German and combined PDFs were rebuilt from the synchronized v2 source set. Version 2.0 (May 2026) Source audit: internal Zenodo references were checked against the Zenodo API and moved to stable Concept-DOIs where appropriate. Privacy / publication hygiene: a public plaintext email address was removed from the paper title page and replaced by ORCID. Bilingual release: the live file set now includes English, German, and combined PDFs. Metadata completion: creator affiliation, language, visible draft notice, keywords, related identifiers, and concrete GitHub project path are set. PDF quality: active logs are free of LaTeX errors, undefined references/citations, true rerun warnings, and overfull boxes; 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.

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