Assume each child has at most M/8 wires. An active depth-d call then satisfies H<M<=Cs 8^(-d), so d<=log2(lambda)/6+O_C(1). Consequently u=Theta_C(lambda/2^d); for sufficiently large s this is at least lambda^(4/5) and at most M. The cumulative sum gives a<=4epsilon lambda. Hence L=2^a and D_arr=2^(12a) are both at most s^(1/1000), eventually. These estimates are conditional on shrinkage until the child construction closes the induction. The eventual threshold may be enormous; failure at a practical size does not contradict the asymptotic inequalities.