Predictable Mean-Field Chaos in Random Recurrent Neural Networks
Dynamical mean-field theory (DMFT) maps deterministic chaos in random recurrent neural networks to an effective Gaussian process, usually treated as an ensemble description rather than a predictor of individual trajectories. Contrary to this view, we show that the chaotic mean-field process of the Sompolinsky--Crisanti--Sommers model can be perfectly predictable at the level of a single neuron. When the nonlinearity has a Gaussian or faster Fourier decay, the exact continuous past of one realization determines its entire future: the conditional prediction error vanishes despite a positive Lyapunov exponent. We trace this result to complex-time singularities of the self-consistent covariance through the Paley--Wiener criterion. A Lanczos/Krylov representation turns the covariance into a state-space predictor and defines $\alpha$, the rate at which predictive information is transferred to higher temporal modes. For smooth odd nonlinearities in this predictable class, the Krylov rate and the largest Lyapunov exponent scale differently near the chaotic transition, demonstrating that predictive complexity and microscopic instability are distinct characteristics of the dynamics. Resolving the first $p$ Krylov modes yields a prediction horizon that grows as $\log p$ and enables finite-network forecasts from a single neuron's observed past, without knowing the connectivity and without observing the rest of the network. Thus, the mean-field power spectrum becomes a diagnostic of whether apparent variability reflects irreducible fluctuations or hidden deterministic structure encoded in the observed past.