Negative-Entropy Construction Theory II

This paper develops a geometric and dynamical extension of the logical-domain framework established in Negative-Entropy Construction Theory I. Rather than assuming that a logical domain necessarily possesses a manifold structure or an intrinsic dynamics, we introduce the additional structures required to construct such realizations. We begin with an environment configuration space and define the feasible state set through the relations and constraints of a logical domain. We then introduce admissible transition relations, local admissible variations, and the distinction between geometric degrees of freedom and dynamically permitted degrees of freedom. Under appropriate regularity conditions, the feasible state set admits a manifold realization, while its tangent spaces describe geometrically admissible directions. We further examine the influence of global topology on vector-field existence and dynamical behavior, including the Poincaré–Hopf theorem, the Poincaré–Bendixson theorem, and the distinction between low-dimensional and higher-dimensional autonomous dynamics. Smooth, nonsmooth, and hybrid dynamical systems are considered as different realization mechanisms for admissible evolution. Finally, we define dynamical classification through invariant sets, attractors, basins of attraction, and asymptotic behavior. We establish sufficient conditions under which finitely many basins of attraction form a complete partition of the feasible state space, while explicitly allowing for unclassified regions in the general case. The resulting framework provides a structured passage from logical-domain constraints to feasible configurations, local freedom, geometric realization, and dynamical classification. It does not claim that every logical domain necessarily generates a manifold or a dynamical system.