Functional Stability Theory — A Programmatic Hub: Universal Convexity-Uniqueness Across Scales (From Particles to Cosmology)

Functional Stability Theory (FST) is a research programme that proposes a single stability principle — the Universal Convexity-Uniqueness criterion (UCU) — as a unifying framework for emergent structure across physical and mathematical scales. Version 1.6 adds an operative claim-gates table that distinguishes current support, next upgrade gate and revision signal for each programme line. It also aligns the UCU status map and AI disclosure with this gate-ledger discipline so local or conditional results are not read as proof upgrades. This article is the programmatic hub of the FST series. It synthesises the four application-scale companions (FST-I through FST-IV) under a common architecture: universal game + MEPP axiom + UCU three-lemma endgame (UCU1 strict convexity, UCU2 existence, UCU3 quadratic lower bound) + RG bridge between scales. FST Series at a Glance SlotScaleTitleConcept-DOI FST-IParticles / ThermodynamicsA Game-Theoretic Framework for the Thermodynamic Stability of Fundamental Parameters10.5281/zenodo.20130544 FST-IIChemistry / AutocatalysisChemical Stability and Autocatalytic Selection10.5281/zenodo.20130563 FST-IIIBiology / Protein foldingBiological Stability and Nash Frustration10.5281/zenodo.20130573 FST-IVCosmology (collector slot)Cosmological Stability — Collector for CRM-Series and Dark Energy (forthcoming)— ↳ FST-DE↳ sub: Dark EnergyDark Energy as Residual Vacuum Free Energy10.5281/zenodo.19036235 HubProgramme umbrellathis paper10.5281/zenodo.20130499 FST Foundations (cited building blocks) The FST programme rests on several published foundation papers in the mathematics and physics lines. The Hub references them; the full list with DOIs: Foundation PaperRoleConcept-DOI RH — Even Dominancespectral-stability prototype10.5281/zenodo.19764771 Zeta Zooclassification framework10.5281/zenodo.19673226 Zookeeper (CCM-Route)RH technical core10.5281/zenodo.19673126 FST Spectrum DualityPhysics-Master (RFEP)10.5281/zenodo.19036190 Hodge PositivityAP / algebraicity, Pattern-A lesson source10.5281/zenodo.19087439 BSD PositivityNéron–Tate height architecture10.5281/zenodo.19087443 P vs NP Entropycomplexity reformulation10.5281/zenodo.19056809 Yang–Mills Mass Gapstrong-coupling lattice route10.5281/zenodo.19087433 Navier–Stokes (Attractor)3D NS regularity framework10.5281/zenodo.19087449 NS-LDITLL/LDI bridge10.5281/zenodo.19056807 K41 Variational Minimiserunconditional Kolmogorov joint-minimiser theorem10.5281/zenodo.20131305 Turbulence / DFC cascade routeconditional anomalous-dissipation companion10.5281/zenodo.19056813 Honest Status The paper is an honest survey: it describes what the programme delivers, what remains conditional, and which sub-problems are open (HD3' for Beal, full UCU3 for Yang–Mills, L4 RG-matching for the dark-energy/CRM-VI line). It does not prove or enlarge any individual hard open problem; FST fixes the shape of the conditional statement, and the application-specific UCU3-global step in each hard case is identified with the original difficulty. Code & sources GitHub: research-line/functional-stability-theory. Hub LaTeX sources and compiled PDFs are at applications/fst-hub/. Building-block papers (RH, YM, NS, K41, Turbulence/DFC, Hodge, BSD, P vs NP, Zookeeper, Zeta Zoo, Spectrum Duality) live in their respective fst-mathematics/, fst-physics/, masters/ sub-paths. Changelog Version 1.7 (June 2026) Citation metadata alignment: The RH foundation reference now consistently points to the RH Even Dominance Concept DOI 10.5281/zenodo.19764771, matching the project-local citation check and the in-place Zenodo metadata correction already applied to v1.6. German terminology maintenance: The German PDF incorporates the June 2026 terminology and style pass, including tighter scale-, framework- and gate-language while preserving real German umlauts in the visible text layer. No claim upgrade: This is a maintenance release for identifier, terminology and artifact parity. It does not prove or enlarge any FST component theorem and keeps the Hub's survey/status-ledger role unchanged. Verification: English, German and combined PDFs were checked as a three-file release set, including PDF metadata, file hashes, log scan and German text-layer umlauts. DE/EN: English and German PDFs are published synchronously; the combined PDF is generated from EN plus DE. Version 1.6 (June 2026) Gate ledger: Added an operative claim-gates table distinguishing current support, next upgrade gate and revision signal for each programme line. Claim hygiene: The UCU status map is now read together with explicit upgrade conditions, preventing local or conditional results from being read as proof upgrades. AI disclosure alignment: The disclosure text now includes the gate-ledger check as part of programme-level consistency review. Verification: English, German and combined PDFs were rebuilt and checked; German sources preserve real umlauts. DE/EN: English and German PDFs are published synchronously; the combined PDF is generated from EN plus DE. Version 1.5 (May 2026) Citation hygiene: previously unreferenced bibliography items for Wilson--Kogut, Jacobson, Maynard Smith, the unified FST overview and the CRM line are now cited in the relevant conceptual sections. Bibliography metadata: Wilson--Kogut, Zeta Zoo, RH/Even-Dominance and Yang--Mills labels were aligned with current Crossref/Zenodo metadata while preserving Concept-DOI references. Series alignment: the Turbulence / DFC cascade route is now explicitly represented in the paper body as the Paper-B companion to K41, matching the already-live Zenodo metadata entry 10.5281/zenodo.19056813. Policy hygiene: obsolete AI-acknowledgement wording was removed; AI use remains documented only in the neutral AI Disclosure section. Verification: EN/DE cite-key and bibliography sets are balanced at 25/25; logs are free of citation, undefined-reference, rerun and overfull warnings. DE/EN: German and English PDFs were rebuilt together, and the combined PDF was regenerated from the fresh language PDFs. Version 1.4 (May 2026) Source audit: all cited FST Foundation Zenodo identifiers were checked against the Zenodo API/DOI resolver. Critical metadata correction: stale RFEP/Spectrum Duality DOI 10.5281/zenodo.19672198 was removed and replaced by Concept-DOI 10.5281/zenodo.19036190. Persistent identifiers: RH, Zeta Zoo, Zookeeper, Hodge and CRM references were shifted from version-specific records to Concept-DOIs where appropriate. Bibliography: external standard references now include verified DOI resolver links. Preprint marking: visible draft/preprint note added to the Description and Notes. Version 1.3 (May 2026) Title: triple-hyphen --- replaced by Unicode em-dash — (proper typographic rendering on Zenodo). Description rewrite: duplicated changelog section removed; consolidated to a single Changelog block. New "FST Series at a Glance" table with all four numbered slots (I/II/III/IV) plus Hub and FST-DE sub-component. New "FST Foundations" table listing the twelve cited building blocks with persistent DOIs, now separating K41 Paper A from the Turbulence/DFC Paper B companion. GitHub note added. Related Identifier corrected: FST-DE entry relabelled to "FST-IV Cosmology / FST-DE Dark Energy" to reflect the structural positioning established in v1.2. Version 1.2 (May 2026) Structural clarification: FST-IV is the cosmological collector slot (Plan.txt Level-2 scale). FST-DE is a sub-component of FST-IV (the dark-energy chapter). The Hub paper retains its no-numeral position as the programmatic umbrella. Version 1.1 (May 2026) Title change: former "Functional Stability Theory IV: An Overview" renamed to "Functional Stability Theory — A Programmatic Hub: Universal Convexity-Uniqueness Across Scales". The Roman numeral "IV" was removed because the hub paper is positioned as a series umbrella, not as a chronological fourth instalment. Bibliography updated with FST-I/II/III Concept-DOIs. Version stamp removed from title page (per LATEX_PDF_CHECKLISTE). Version 1.0 (May 2026) Initial release as part of the FST series. Hub paper with cross-links to FST-I/II/III companions and references to the published FST mathematics and physics building blocks.