Isolation of Topological Degrees of Freedom under Coarse-Graining

This work presents an exact finite-dimensional construction for isolating topological degrees of freedom under coarse-graining in a two-dimensional Z2 lattice gauge theory on a finite torus. The construction combines cellular homology and cohomology, gauge-quotient physical configuration spaces, tree-cotree polarization, finite-dimensional Fourier duality, and independently constructed coarse-graining isometries. The main result is an exact intertwining relation between the microscopic duality transformation and the effective transformation induced on the retained topological Hilbert space. The work is accompanied by a computational verification implementation using exact arithmetic over GF(2). The results constitute a controlled finite-dimensional benchmark and do not claim universality for arbitrary gauge theories, higher genus, non-Abelian theories, or generic renormalization-group constructions.