Observer-Coupled Viability in Universal Viability Systems: Epistemic Contraction, Phase Transition, and the Joint Fixed-Point Theorem

Papers P1 through P8 of the Karimov–Alekberli (KA) Framework established the Universal Viability System (UVS) and proved its Closure Theorem under a structural assumption that remained implicit: the observer measuring the system is epistemically external and dynamically inert. This paper makes that assumption explicit, formally discloses it as the boundary condition of P8, and then removes it. We introduce the Observer Viability Functional PhiO (t) as a first-class dynamical object governed by the same UVS axioms as the system it observes. A bounded coupling operator K connects system and observer dynamics, inducing an epistemic permeability coefficient Psi (K) that modulates joint viability. Three coupling regimes are characterized: decoupled (recovering P8), weakly coupled (additive approximation), and strongly coupled (non-separable multiplicative factorization). The central result — the Epistemic Contraction Theorem — establishes that the observer viability iteration is a contraction mapping if Psi (K) exceeds a critical threshold Psic determined by the system viability signature — a sufficient condition for existence of a unique observer fixed-point. Below this threshold, no fixed-point exists in the admissible Banach space: the epistemic collapse regime. The Joint Fixed-Point Theorem characterizes theunique equilibrium (PhiS*, PhiO*) in the viable region Rᵥiable = Psi (K) > Psic, PhiS > 0, PhiO > 0 and identifies the epistemic phase transition surface at its boundary. P8 is recovered as the special case Psi (K) = 1, PhiO = 1. The framework reinterprets P1–P8 as first-order projections of a second-order theory and positions UVS as a two-system epistemic stability geometry rather than a single-system viability calculus.