SPECTRAL EMERGENCE IN SUPERPOSED LATENT REASONING: PHASE-COHERENT BRANCH TRANSMISSION FROM INPUT TO ANSWER

Continuous chain-of-thought replaces decoded intermediate tokens by a sequence of latent states whose superposition can carry several reasoning branches at once. We study how those branches evolve as dynamical signals and how their oscillatory structure becomes an answer. Each branch coefficient is decomposed into reasoning-time frequencies, with amplitude, frequency, damping, and phase describing its strength and temporal alignment. The answer readout then acts on the branch vibrations through a readout Gram matrix: constructive and destructive phase coherence determines which branches help, suppress, or cancel an answer. We derive a branch-resolved spectral attribution theorem and an answer-relevant marginal utility that predicts branch-removal and phase-shift interventions. The input–state–output response supplies the propagation bridge and identifies the excitation, propagation, and visibility factors; finite-window DFT is used only as a numerical estimator of the underlying frequency response. The controlled path-compositional suite audits the identities and intervention scores, while the theory states the conditions under which the spectral quantities can be interpreted as reasoning mechanisms.