Theorem 19.6

Theorem 19.6 (The linear access certificate is pointwise sharp). For every horizon $N\ge0$, some finite controlled-Dyck presentation attains the displayed upper bound. For every $N\ge1$, one can arrange $B_M=2$ and $$a_{L_M}(N)=3N+2=(N+1)B_M+N.$$ The presentation is allowed to depend on $N$.