Proposition 7.1

Proposition 7.1 (Future transition after type birth). For $r\geq1$, put $A_r=0^{2r}$, $$P_r=A_r11A_r, \qquad T_r=1A_r1.$$ Then $D_1(P_r)=A_r$, so the transition selected by $1$ has target $T_r=1D_1(P_r)1$. The type $P_r$ is born by the end of the prefix $P_r$, but $T_r$ is not a factor of that prefix. It becomes available only in a later history such as $P_r1P_r$. Thus the immutable first-birth record of $P_r$ cannot already point to every future transition out of $P_r$.