Symmetry-protected delay spectroscopy in oscillator networks

We study how path-specific delays can be identified from frequency-response measurements in delay-coupled oscillator networks. Although time delays strongly shape collective dynamics, measured transfer curves usually combine many directed routes, making it difficult to determine which delayed path controls a chosen source-detector channel. We show that discrete symmetry can resolve this inverse problem. For delay-coupled Kuramoto populations on a locked Ott-Antonsen branch, a symmetry-preserving operating point enforces an exact detector-source response zero. A controlled symmetry-breaking detuning unfolds this protected zero into a ladder of real-frequency nodal crossings. We prove a general theorem for finite retarded delay networks, solve the minimal four-population motif explicitly, and show that the asymptotic spacing of the nodal ladder reads out a detector-selected delay. In the baseline motif, the recovered spacing matches the predicted delay within 0.08%, while higher-node validations recover effective delays of 2.5027 and 2.7030, in close agreement with the corresponding microscopic path delays. The result provides a swept-frequency protocol for symmetry-assisted delay spectroscopy, requiring only a selected linear detector-source response rather than reconstruction of the full transfer matrix.