Papers P9 and P10 of the Karimov-Alekberli (KA) Framework established observer viability existence (P9) and observer phase multiplicity (P10), treating the system viability PhiS as an external signal. The present paper removes this asymmetry and studies the fully coupled dynamics in which PhiS and PhiO co-evolve under mutual coupling. The central object is the equilibrium manifold E (kappa) = (PhiS*, PhiO*): 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 Kbif. Second, the Joint Bifurcation Classification: the manifold singularities correspond to exactly four structurally stable bifurcation types in the (PhiS, PhiO, 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 (PhiS (t), PhiO (t) ) exhibits path dependence that is strictly stronger than P10's observer-only hysteresis -- closed loops in (PhiS, PhiO) 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.
Karimov et al. (Mon,) studied this question.