Theorem 19.9

Theorem 19.9 (Dyck-trace minimax certificate). For every controlled-Dyck language inside the fixed envelope $D$, $$a_L(N)\le\max\{1,(N+1)\beta_D(L)+N\}.$$ The number $\beta_D(L)$ depends only on $L$ and the selected Dyck envelope, not on one chosen regular controller.