What question did this study set out to answer?

This work aims to present Functional Stability Theory as a framework linking number theory, physics, and cosmology.

June 21, 2026Open Access

Functional Stability Theory — Physical Instantiation via the Renormalized Free Energy Principle

Key Points

This work aims to present Functional Stability Theory as a framework linking number theory, physics, and cosmology.
Introduced the physical instantiation of Functional Stability Theory (FST) via the Renormalized Free Energy Principle (RFEP).
Outlined ontological layers including abstract rules and domain-specific verification targets.
Updated the framework to adhere to strict-review and source-check protocols.
Identified specific conditions for domain transfer claims, emphasizing the necessity for independent certification.
Reframed the role of RFEP as a stability meta-rule rather than a closure theorem.
Clarified the critical gaps in the framework related to mass-defined effective diagnostics.

Abstract

We present the physical master paper of Functional Stability Theory (FST): a framework that treats functional positivity under a gauge constraint as a shared structural challenge across number theory, mathematical physics, and cosmology. This v1. 8 release keeps the live v1. 7 Zenodo description design with the Five-Masters series structure, but updates the RFEP paper to the 2026-05-20 strict-review guardrail and the 2026-05-26 source-check refresh. Three ontological layers are distinguished: (1) the abstract RFEP as an organising stability meta-rule; (2) the Dissipative Selection Principle (DS1-DS3) ; and (3) domain-specific instantiations as verification targets rather than completed domain proofs. The v1. 8 text explicitly treats Zookeeper-MS2/C2 as a conditional microcluster normal form with open uniform gates and requires independent target-subspace or outer-gap certificates before a domain transfer can be read as more than posterior self-consistency. The Five Masters Zookeeper: The Spectral Zookeeper, Concept DOI 10. 5281/zenodo. 19673126. Zeta Zoo: Branch-Local Hilbert-Polya, Concept DOI 10. 5281/zenodo. 19673226. Dirichlet Atlas: Zoo-Mapping / character atlas, Concept DOI 10. 5281/zenodo. 19960809. Selberg: NE-B failure as Hilbert-Polya detection, Concept DOI 10. 5281/zenodo. 19962588. RFEP: this physical master paper, Concept DOI 10. 5281/zenodo. 19036190. CHANGELOG Changes in Version 1. 9 (June 2026) Maintenance release candidate after the post-v1. 8 certificate-ledger guardrail and English-style pass. The release keeps RFEP as a framework and transfer-audit layer; it does not claim a completed domain proof for Yang-Mills, Navier-Stokes, turbulence, cosmology, or RH. Certificate-ledger guardrail: the post-v1. 8 text makes independent target-subspace, outer-gap, spectral-exactness, detector-degree, advice-budget, and repair-certificate gates more explicit before any domain-transfer claim can be strengthened. Claim hygiene: the RFEP layer is framed as an organising stability meta-rule and diagnostic framework, not as an automatic closure theorem for the domain papers. Style maintenance: the English register was tightened and idiomatic wording was smoothed after the June 2026 style pass. PDF hygiene: the English, German, and combined PDFs were rebuilt from the active v0. 9 source set; German PDF text extraction uses real umlauts. DE/EN: English and German PDFs remain synchronized in the local candidate set; the combined PDF merges English first, then German. Changes in Version 1. 8 (May 2026) Critical: Reframed the Zookeeper import as a conditional microcluster normal form with explicit open uniform gates sₗambda -> 0, pₗambda -> 0, and g_* >= g0 > 0. Critical: Demoted the FST-7 equivalence language to a conditional compatibility schema and marked mass-defined effective gaps as posterior diagnostics unless independently certified. Major: Reorganised RFEP as the physical instantiation of FST rather than an appendix-like spectrum-zero route map; added branch locality, Zeta-Zoo separation, physics/cosmology proof-status mapping, and the CRM loop in the outlook. Major: Corrected the C2ca. 5 Laplace-gap justification: the values g_* approx 5--7 are numerical diagnostics of the full mass-defined cluster gap, not consequences of the rough potential-free lower bound. Minor: Removed the self-reference Bibitem RFEPFramework, aligned Zookeeper/Zeta-Zoo references with stable Concept DOIs, and anchored eight previously uncited bibliography entries in the EN/DE text. Source check: Corrected Connes 2026, CCM 2024, RH Landscape/Even-Dominance, Cornelissen--Marcolli 2010, and several DOI metadata fields against arXiv, DOI/Crossref, AMS, and Zenodo API records. DE/EN: English, German, and combined PDFs were rebuilt; the combined PDF follows the project convention English first, then German. Technical info Public GitHub project path: research-line/functional-stability-theory/tree/main/masters/spectrum-duality. Published in English and German versions; the combined PDF merges the two language versions. Keywords: Functional Stability Theory, RFEP, Pattern A, Dissipative Selection, Gauge Constraint, Spectrum Duality, Yang-Mills, Navier-Stokes, Dark Energy, CRM, Renormalized Free Energy

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