Characterizing universal topological phases beyond spatial embedding

Although Hamiltonians govern wave-matter interactions, conventional characterizations of topological phases of matter require more than the Hamiltonian alone, relying on global invariants defined on closed reciprocal-space manifolds or local commutation relations with position operators. This spatial dependence precludes a unique characterization of topological phases on an abstract platform, whose Hamiltonian can be defined independently of its spatial embedding. Here we establish a framework for characterizing universal topological phases, grounded in the invariance of topological features under arbitrary Hamiltonian-preserving deformations. We construct a Hamiltonian-based spatial embedding that groups a system's constituents according to their connectivity. This embedding, which reproduces the results of conventional topological diagnostics in established systems, enables a coherent extension of topological phases to complex networks using the Hamiltonian alone, further highlighting exceptional robustness in small-world networks. Our results reshape the conceptual foundations of topological physics by decoupling topological characterization from prescribed physical platforms.