Topological phases of matter exhibit robustness, quantized response, and protected boundary modes that cannot be explained by local symmetry breaking or microscopic Hamiltonian details alone. We present a structural account of topological phases based on coarse-grained projection and global admissibility of local effective descriptions. Phases correspond to admissible basins of projected dynamics, and topological phases arise precisely when locally well-defined effective descriptions fail to admit a globally consistent completion. Topological invariants appear as obstruction classes of overlap structures, explaining quantization and robustness. Bulk–boundary correspondence follows inevitably: protected boundary modes arise as compensating degrees of freedom required to restore local consistency at interfaces. The framework applies to interacting and disordered systems with a spectral gap or mobility gap and does not rely on band topology or free-fermion assumptions. All results are formulated at the level of effective descriptions.
Peter Nero (Thu,) studied this question.