Joint Viability Geometry and Reflexive Hysteresis: Topology of the Equilibrium Manifold in Coupled System-Observer Dynamics

Papers P9 and P10 of the Karimov-Alekberli (KA) Framework established observer viability existence (P9) and observer phase multiplicity (P10), treating the system viability Phi_S as an external signal. The present paper removes this asymmetry and studies the fully coupled dynamics in which Phi_S and Phi_O co-evolve under mutual coupling. The central object is the equilibrium manifold E(kappa) = {(Phi_S*, Phi_O*) : joint fixed-point conditions hold}, whose topology as a function of the coupling parameter kappa characterizes all qualitatively distinct regimes of the coupled system.

We establish four main results. First, the Equilibrium Manifold Structure Theorem: E(kappa) is a smooth manifold for generic kappa, with singularities (folds, cusps) at critical values K_bif. Second, the Joint Bifurcation Classification: the manifold singularities correspond to exactly four structurally stable bifurcation types in the (Phi_S, Phi_O, kappa) parameter space -- saddle-node (fold), cusp catastrophe, and -- under additional symmetry or boundary conditions -- transcritical and Neimark-Sacker (oscillatory) bifurcations. Fold and cusp are generic (guaranteed by J1-J4); transcritical and Neimark-Sacker are conditional (requiring additional hypotheses J5 and NS respectively). Third, the Reflexive Hysteresis Theorem: under strong mutual coupling, the trajectory of (Phi_S(t), Phi_O(t)) exhibits path dependence that is strictly stronger than P10's observer-only hysteresis -- closed loops in (Phi_S, Phi_O) state space are possible, corresponding to reflexive cycles in which observer and system mutually reinforce degradation or recovery. Fourth, the Topological Obstruction Theorem:certain transitions between qualitatively distinct joint equilibria are topologically forbidden -- the equilibrium manifold cannot be continuously deformed from one regime to another without passing through a bifurcation.

Together, these results complete the transition of UVS from a viability calculus to a viability phase geometry: the primary object is no longer a scalar functional Phi but the topology of the set of joint equilibria and the bifurcation events that change it.