Size Knowledge Accelerates One-Bit Las Vegas Broadcast in Anonymous Dynamic Networks

We study the running-time value of network-size knowledge for Bernoulli Reactivation Flooding (BRF), a two-state private-coin protocol for idle-start stabilizing broadcast in anonymous, port-indistinguishable, synchronous, 1-interval-connected dynamic networks. For fair-coin BRF, the causal state-adaptive worst-case expected stabilization time is 2Θ(n2)2^{\\Theta(n^2)}. We show that when the network size nn is known, the same two persistent protocol states suffice for expected stabilization time 2Θ(nlog⁡n)2^{\\Theta(n\\log n)} by choosing a size-aware dyadic drop probability. We also prove that for every fixed survival probability, a causal state-adaptive topology adversary maximizes the Bellman continuation value by exposing exactly one idle frontier node whenever the active set is a proper subset of the network. Finally, we prove that 2Θ(nlog⁡n)2^{\\Theta(n\\log n)} is asymptotically optimal over the homogeneous fixed-bias BRF family against causal state-adaptive adversaries. The optimality claim is specific to the homogeneous fixed-bias BRF family and is not a lower bound for all two-state randomized broadcast protocols. The accompanying repository provides the LaTeX source, exact-rational Bellman experiments, high-precision numerical evaluations, figures, and audit materials. Preprint; not peer reviewed.