A nested pinched center frontier has a strict inner-disk search

The given inner disk $K$ is bounded by the simple directed inner homoclinic loop and occupies one saddle quadrant at $q$. These are exactly the hypotheses of [F1], so $c_K-s_K=1$ and there is a center $c$ strictly inside $K$. The boundary saddle $q$ is not counted. No half-branch classification is needed for this application.

The selected basin center does not lie in $K$ by hypothesis, so the center $c$ found in step 1.1 is a center different from the selected one; the search that starts from $c$ is therefore a search in a strictly smaller region of the source disk.

The boundary $\mathrm{\partial }K$ is a union of trajectories, hence invariant; by uniqueness of trajectories [F3] no regular trajectory crosses it. Consequently every nested periodic disk around $c$ lies inside $K$: its frontier cannot reach the exterior of $K$ without crossing $\mathrm{\partial }K$, and if its frontier reaches $\mathrm{\partial }K$ it must coincide with one of the two boundary circuits, which is the original single circuit rather than a new nested two-loop frontier.

Let a new nested two-loop frontier occur in the search from $c$; by step 3.1 its two loops, and in particular its saddle $r$, lie in $K$, and it cannot be the original frontier, so $r$ is strictly interior to $K$. Its inner disk $K^{\mathrm{\prime }}$ is bounded by its inner loop and, by [F2], is the bounded component of the complement of that loop; since the loop lies in the interior region swept by the search and $K^{\mathrm{\prime }}$ is the smaller bounded side, $K^{\mathrm{\prime }}\subseteq K$ and $r\mathrm{\notin }{K}^{\mathrm{\prime }}$. Every saddle interior to $K^{\mathrm{\prime }}$ is then interior to $K$, while the interior saddle $r$ of $K$ is not available inside $K^{\mathrm{\prime }}$; hence the number of interior saddles of $K^{\mathrm{\prime }}$ is strictly smaller than that of $K$.

Collecting steps 1.1-4.1, a search from any center of $K$ remains inside $K$, and every nested two-loop frontier encountered has an inner disk properly contained in $K$ with strictly fewer interior saddles; no essential boundary class was used or assigned to $K$, and only the two cited suppliers and the local uniqueness of trajectories were consumed.