Topological Phases as Bundle and Index Data: A Conditional Projection-Admissible Encoding in Modal Triplet Theory

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.