The Phase Is the Gradient: Equilibrium Propagation for Frequency Learning in Kuramoto Networks

We prove that in a coupled Kuramoto oscillator network at stable equilibrium, the physical phase displacement under weak output nudging is the gradient of the loss with respect to natural frequencies, with equality as the nudging strength beta tends to zero. Prior oscillator equilibrium propagation work explicitly set aside natural frequency as a learnable parameter; we show that on sparse layered architectures, frequency learning outperforms coupling-weight learning among converged seeds (96.0% vs. 83.3% at matched parameter counts, p = 1.8e-12). The approximately 50% convergence failure rate under random initialization is a loss-landscape property, not a gradient error; topology-aware spectral seeding eliminates it in all settings tested (46/100 to 100/100 seeds on the primary task; 50/50 on a second task, K-only training, and a larger architecture).