Optimal exponential memory for sequential Euclidean connections: edge-power costs and phase transitions

We study the edge-power cost of the labelled tree generated by the $γ$-strategy, a constant-gain rule for sequential Euclidean connections. Starting with $x_0=p_0$, each input point $p_i$ is attached to $x_{i-1}$, and the state is updated by $x_i=γx_{i-1}+(1-γ)p_i$. Retaining $x_i$ subdivides the in