Common plaque lifted caps admit nested source-disk inclusions

Fix a late cap $C_n$ and a later cap $C_m$ from the recurrent family. The projected image of $C_n$ is compact and disjoint from the original loop $\gamma$ by the avoidance clause of [F1], so the two compact sets $\mathrm{image}(C_n)$ and $\gamma (S^1)$ have positive distance by [F2]; uniform convergence ${\gamma }_{t(s)}\to \gamma$ therefore implies that every sufficiently late boundary ${\gamma }_{t(s_m)}$ avoids the entire projected image of $C_n$. Base both lifted caps in the universal cover of $B$ at the same point of their common interior plaque patch.

The Jordan disk regions ${\mathrm{\Delta }}_n$ and ${\mathrm{\Delta }}_m$ overlap at that common point, and $\mathrm{\partial }{\mathrm{\Delta }}_m$ is disjoint from ${\mathrm{\Delta }}_n$ because its projection avoids the image of $C_n$. Any path in the connected disk ${\mathrm{\Delta }}_n$ from the common interior point to another point cannot exit ${\mathrm{\Delta }}_m$ without crossing $\mathrm{\partial }{\mathrm{\Delta }}_m$, so ${\mathrm{\Delta }}_n\subseteq \mathrm{int}{\mathrm{\Delta }}_m$. The disk parametrizations into ${\mathrm{\Delta }}_n$ and ${\mathrm{\Delta }}_m$ are diffeomorphisms by [F1], so their inverses compose to a $C^2$ embedding $h_{n,m}:D\to \mathrm{int}D$ on the actual reference C² disk region $D$ of the development satisfying $C_n=C_m\circ h_{n,m}$ as projected maps.

Fix the recurrence sequence once. At each stage take the least later index whose boundary avoids the preceding compact cap image; step 1.1 guarantees such an index. This is a deterministic recursion on natural numbers, requiring no dependent choice, and produces the required increasing sequence of source-disk inclusions whose projected cap maps agree on the included disks. This is nesting of source disks in one based universal cover, not a claim that the ambient projected images are embedded disks, and the maps $h_{n,m}$ are exactly the base gluing maps needed by the immersed paired-sweep seam lemma.