Mathematical Implications of Non-Homogeneous Para- Topological Convergences Between Orthogonal Unbridged n- Spaces III

Developed is an effective mathematical framework for a hypothetical crossing between twootherwise orthogonal, non-bridged (n)-dimensional spaces. The crossing is represented by alocalized interaction kernel with finite radial extent and finite transverse thickness. Derived then areeffective equilibrium conditions for the crossing radius and thickness, investigated is the associatedgeometric screening of relative modes, and analyzed is the stability of an induced tachyonic sector.The model admits a parameter regime in which the crossing possesses a finite stationary radius,positive radial and transverse stiffness, strong suppression of relative geometric modes, and apositive effective mass squared for the otherwise tachyonic sector. The resulting configuration isweakly gravitating and does not approach a horizon-forming compactness regime within thereduced model.Further introduced is an effective entropy-transfer channel between the two sectors. The analysisshows that strong geometric screening can suppress thermodynamic exchange without requiringcomplete geometric decoupling. A representative benchmark therefore exhibits the simultaneousproperties of finite crossing, geometric screening, tachyon quenching, and approximatethermodynamic isolation. These results constitute consistency conditions of an effective model anddo not establish the physical existence of inter-universe crossings.