Proposition 7.3

Proposition 7.3 (Bridge-chasing escape). For $r\geq1$ and $d\geq0$, let $$P_{r,d}=A_r(11A_r)^d, \qquad A_r=0^{2r}.$$ These are even palindromes. For $d>0$, $$\lambda(P_{r,d})=P_{r,d-1} \quad\text{and}\quad b_{P_{r,d}}=1,$$ whereas the bridge of $A_r$ is $0$. The search for the $0$-transition from $P_{r,d}$ therefore makes exactly $d$ failed bridge tests before reaching the variable target $A_r$.