A walk-sum framework of frequency-dependent brain communication architecture

Brain oscillations organise neural communication, yet why specific frequencies couple to specific spatial modes remains analytically unresolved. The walk-sum algebra of the structural connectome determines a frequency-dependent transfer function, the resolvent, whose spatial structure follows entirely from topology. With zero free parameters, the bare resolvent predicts a parcellation-invariant crossover near 12.6 Hz, an eigenmodel correlation of $\rho =0.965$ , and five testable spatial predictions. These are confirmed in source-reconstructed MEG from 912 subjects across three datasets and intracranial EEG from 90 epilepsy patients, ruling out volume conduction. A two-parameter dressed resolvent improves prediction; a neural mass negative control $(\rho \approx 0.006)$ confirms the resolvent describes channels, not dynamics. Propofol anaesthesia collapses alpha channels; in schizophrenia, weakened local dynamics expose the structural scaffold-topological transparency. This framework provides the first analytical derivation of frequency-band communication architecture from connectome topology.