arXiv:2604.20369v2 Announce Type: replace Abstract: We study finite-horizon rate-cost tradeoffs for general stochastic control systems, including nonlinear systems. Under an average control cost constraint, we show that the minimum expected communication rate lies within an additive logarithmic gap above the normalized directed information minimized over causal control policies. This establishes directed information as the operationally relevant quantity governing rate-limited control, extending its previously established roles in causal source coding and linear quadratic Gaussian (LQG) control to general nonlinear control systems.