Functional Stability Theory IV: A Scale-Invariant Stability Principle from Particles to Cosmology -- An Overview of the FST Programme

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. This article is the programmatic hub of the FST series. It synthesises the FST-I (Thermodynamic Stability), FST-II (Chemical Stability), FST-III (Biological Stability) and FST-DE (Dark Energy) companion papers 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. The paper is an honest survey: it tabulates which sub-problems are closed and which remain open (HD3' for Beal, full UCU3 for Yang-Mills, L4 RG-matching for the CRM-VI dark-energy line) and records the Pattern A methodological lesson learned from the cosmological route -- a positivity reformulation of an open conjecture is in general only a translation of language, not progress, unless its proof strength is independently verified. CHANGELOG 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 (RH, Zookeeper, Spectrum Duality, Hodge, BSD, P vs NP, Yang-Mills, Navier-Stokes, K41). This is the v1 "honest survey" stage. Future versions (v1.1, v1.2, ...) will tighten cross-references once FST-I/II/III companions receive their own DOIs, and may extend the bibliography and status maps as the programme matures.