Critical Dynamics of AI Self-Improvement

AI systems can accelerate AI research without creating a self-amplifying process. We ask when AI-assisted R\&D crosses into recursive criticality, how quickly amplification develops, and what limits its duration. A delayed capability model yields a recursive reproduction number, $\RAI$, equal to realized recursive gain divided by research-frontier hardening. Perturbations are locally damped for $\RAI<1$ and amplified for $\RAI>1$. The threshold is independent of baseline research speed and feedback delay in the minimal model; delay instead controls the amplification rate and imposes a high-throughput limit $\ln \RAI/\tau$. A finite effective frontier makes supercritical episodes self-limiting. Extending the model to multiple research actors yields a network reproduction number given by the spectral radius of a transfer matrix, so a coupled ecosystem can be supercritical even when every actor is individually subcritical. The framework separates ordinary acceleration from a change in dynamical regime and identifies quantities that empirical studies of agentic AI R\&D would need to estimate.