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 square torus. The physical configuration space is defined as the gauge quotient F_phys = C^1 / im(delta_0) with topological kernel H^1(T^2,F_2), dual to H_1 via a perfect homology-cohomology pairing. A tree-cotree polarization provides explicit local/topological coordinates and separates the local boundary sector. Two coarse-graining isometries V_A^P and V_B^{P_} are constructed independently without using the microscopic duality, by fixing complementary local reference states |0_loc> and |+_loc> while retaining the full 4-dimensional topological Hilbert space through independent type-safe identifications J_A and J_B. The microscopic Hadamard Fourier duality maps gauge-invariant orbit states to cycle states and factorizes as local times topological. The induced effective transformation is defined by type-safe transport D_eff = J_B^{-1} D_top^P J_A and satisfies the exact intertwining relation D_mic^P V_A^P = V_B^{P_} D_eff. In adapted coordinates its matrix equals the topological Fourier matrix. The work is accompanied by a reference verification implementation using exact arithmetic over GF(2) for chain identities, ranks, tree-cotree structure and pairings, and complex arithmetic for the intertwining check. L=2 uses exhaustive spanning-tree comparison (31/31), L=3 to L=6 use sampled comparisons (29/29). Full Hilbert-space vectors are checked for L=2,3, compressed physical coordinates for L=4, and coefficient identities for L=5,6 without materializing dense matrices. The results constitute a controlled finite-dimensional benchmark and do not claim universality for arbitrary gauge groups, higher genus surfaces, non-Abelian theories, generic MERA networks, or thermodynamic/continuum renormalization-group flows.