Eventual access is uniform on the trimmed compact inside core

By compatibility of $d$ and [F3], $B$ and $S$ are compact metric subsets of $\mathrm{\Sigma }$, hence closed. Since $B$ contains an open neighborhood of $S$, we have $S\subset {\mathrm{int}}_{\mathrm{\Sigma }}B$. Closedness of $B$ implies $\bar{B}=B$: a point outside $B$ has a metric ball in its open complement and so is not adherent to $B$. Thus $\mathrm{\Gamma }=B\setminus {\mathrm{int}}_{\mathrm{\Sigma }}B$ is closed in $B$, compact by [F3], and contained in $B\setminus S\subset {\mathrm{\Sigma }}^{\circ }$. Also ${\mathrm{\Sigma }}^{\circ }$ is open in $\mathrm{\Sigma }$ because $S$ is closed.

By [F2] and $S\mathrm{\ne }\mathrm{\varnothing }$, $\rho$ is finite and $1$-Lipschitz, hence continuous: $d(x,y)<ϵ$ gives $\mathrm{\mid }\rho (x)-\rho (y)\mathrm{\mid }<ϵ$. It vanishes on $S$. If $x\mathrm{\notin }S$, closedness of $S$ gives a ball of radius $r>0$ about $x$ disjoint from $S$, so $d(x,s)\ge r$ for every $s\in S$ and $\rho (x)\ge r>0$. For any $\eta >0$, $L_{\eta }$ is closed in $B$: at a point with $\rho (x)<\eta \mathrm{/}2$, the Lipschitz bound keeps $\rho <\eta \mathrm{/}2$ throughout a sufficiently small ball. By [F3], $L_{\eta }$ is compact, and it avoids $S$. Its topology inherited from ${\mathrm{\Sigma }}^{\circ }$ equals that inherited from $\mathrm{\Sigma }$, by transitivity of subspace topologies in [F4]; hence it remains intrinsically compact as a subset of ${\mathrm{\Sigma }}^{\circ }$.

Fix $\eta >0$ and let $U_t={\mathrm{\Sigma }}^{\circ }\cap P_t$. The continuity of the inclusion makes $U_t$ open in ${\mathrm{\Sigma }}^{\circ }$. Eventual entry applies to every point of $L_{\eta }\subset {\mathrm{\Sigma }}^{\circ }$, so the indexed family $(U_t{)}_{t\in \mathbb{R}}$ covers $L_{\eta }$. By its compactness and the indexed ambient-cover criterion in [F4], if $L_{\eta }\mathrm{\ne }\mathrm{\varnothing }$ there are finitely many indices $t_0,\dots ,t_m$ whose sets cover it. Put $t_{\eta }=\max \{t_0,\dots ,t_m\}$, which exists by [F5]. Increasingness gives $L_{\eta }\subset P_t$ for every $t\ge t_{\eta }$. If $L_{\eta }=\mathrm{\varnothing }$, take $t_{\eta }=0$ and the same containment is automatic.

If $\mathrm{\Gamma }\mathrm{\ne }\mathrm{\varnothing }$, its compactness and the continuity of $\rho$ allow [F5] to give $\delta ={\min }_{\mathrm{\Gamma }}\rho$ attained at a point of $\mathrm{\Gamma }$. Step 2.1 and $\mathrm{\Gamma }\cap S=\mathrm{\varnothing }$ give $\delta >0$. For $0<\eta <\delta$, we have $\mathrm{\Gamma }\subset L_{\eta }$. Consequently $N_{\eta }={\mathrm{\Sigma }}^{\circ }\cap P_{t_{\eta }}$ contains $\mathrm{\Gamma }$, is open in ${\mathrm{\Sigma }}^{\circ }$ by [F4], and is open in $\mathrm{\Sigma }$ because ${\mathrm{\Sigma }}^{\circ }$ is open. Increasingness gives $N_{\eta }\subset P_t$ for all $t\ge t_{\eta }$. If $\mathrm{\Gamma }=\mathrm{\varnothing }$, take $0<\eta <1$ and $N_{\eta }=\mathrm{\varnothing }$; the neighborhood assertion is vacuous and the trimmed-core containment still follows from step 3.1. This proves both conclusions, with a single neighborhood valid at all later times.