Cosmological Grand Unification via Hierarchical 32-Chingion Contraction Mappings and Local Suspension of Extensionality: Rigorous Invalidation of the Banach-Tarski Paradox inside Finite Recirculating Universes

This study expands the axiomatic constraints of non-equilibrium steady states (NESS) by formalizing a hierarchical dimensional reduction from a 32-dimensional Chingion algebraic space (CH^32) down to the four-dimensional physical spacetime continuum (M^4). By defining the global universe as a compact, connected topological manifold under finite sub-coverings, the divergence of infinite physical quantities is fundamentally eradicated. Within the sub-Planckian singularity domain (Omega_sing), a localized suspension of the axiom of extensionality is introduced. By redefining the well-definedness of selection functions via the intensional equivalences of Homotopy Type Theory (HoTT), Diaconescu's theorem is structurally suspended. Consequently, the emergence of non-measurable sets under the Axiom of Choice (AC) is blocked, rendering the Banach-Tarski paradox physically unrealizable while ensuring the conservation of Liouville measures and Hamiltonian invariants across the macroscopic domain (Omega_macro). Furthermore, by utilizing the Heine-Borel theorem to enforce finite sub-covering topologies, the global stability of the resulting non-equilibrium steady state is rigorously locked via the Perron-Frobenius max-eigenvector equilibrium condition. This closed mathematical architecture confirms that energy undergoes finite cyclical recirculations between microscopic algebraic singularities and macroscopic horizons under a graph-theoretic topological constraint, providing a paradox-free geometric foundation for cosmological grand unification.