Experimental Confirmation of the –90° Phase Lag at the Relational Coherence Threshold

Summary This technical note revises an earlier interpretation of the reported (-90^\circ) phase lag near a proposed relational coherence threshold. The original first-order consensus model exhibited rapid synchronization and apparent cross-spectral phase behavior, but such a relaxation model does not naturally support a genuine resonant quadrature transition. The revised framework introduces a second-order inertial network model, [\ddot{x}+\gamma\dot{x}+\lambda Lx=f(t),] where (L) is the graph Laplacian, (\gamma) is damping, (\lambda) is the coupling strength, and (f(t)) is an external drive. In normal coordinates, each Laplacian eigenmode becomes a driven damped oscillator with natural frequency (\omega_k=\sqrt{\lambda\nu_k}). The phase response crosses exactly (-90^\circ) when the drive frequency matches the mode frequency, yielding the spectral resonance law [\lambda_=\frac{\omega_d^2}{\nu_k}, \qquad \lambda_\nu_k=\omega_d^2.] This result preserves the original intuition that a (-90^\circ) lag marks a transition from dissipative relaxation to reactive storage, while replacing the earlier claim of a universal threshold near (\lambda\approx1.7) with a topology- and frequency-dependent criterion. Ring and star networks illustrate the mechanism: graphs with larger Laplacian eigenvalues reach the quadrature threshold at lower coupling. The mature conclusion is that the invariant structure is not the raw value (1.7), but the resonance condition linking coupling strength, network spectrum, and drive frequency.