On the Structural Boundaries of Disconnection Probability in Finite Lattices and Geometric Convergence to Infinite Limits across Three Major Percolation Models

This paper presents a comprehensive mathematical model examining percolation theorythrough the lens of structural boundaries within finite-sized lattices. While conventionalframeworks focus on proving critical thresholds exclusively in infinite scaling limits, thisstudy defines the discrete physical boundaries for a finite system of linear size a. Specifically,we formulate the minimum configuration required for complete flow blockade (the best-casescenario) and the maximum configuration where a fluid path might still persist (the worstcase scenario) across site, bond, site-bond mixed, and highly connected diagonal hybridmodels. By computing the geometric centers of mass of these dual extreme configurations,we elucidate their behavior in 2D and 3D spaces, demonstrating an exact convergence towardthe theoretical critical probability of 50% as the system expands toward infinity.