Theorem 19.5

Theorem 19.5 (Exact access calculus for controlled Dyck presentations). For every finite controlled-Dyck presentation and every horizon $N$, $$a_{L_M}(N)=\max_{K\in S_N}\alpha_{M,N}(K).$$ The quantity is effectively computable by finite balanced-distance relaxation, the stack recurrence above, and the finite controlled-Dyck shadow quotient. Moreover, $$\boxed{ a_{L_M}(N)\le\max\{1,(N+1)B_M+N\}. }$$