A nonzero limitwise-nullhomotopy class forces a compact boundary leaf

Normalize the arbitrary nonzero $\mathrm{\Pi }$ representative to a short one-field normal fence by [F2], obtaining a finite crossing-word rank; at any positive lifted collision cut into two null factors, retain their actual maximal common closed/null interval and choose an essential lower-endpoint factor; the rank decreases, so after at most $N$ cuts all positive lifts are simple.

Work in the ambient component $M_0$ containing $L$. It is closed, connected and oriented, and inherits the $C^2$ cooriented foliation. The original nonzero $\mathrm{\Pi }$ class therefore excludes every sphere leaf and every sphere universal cover in $M_0$ by [F6]. The original fence and its subword reductions stay in $M_0$, because their connected traces meet that component, which is both open and closed. Thus every positive cap leaf in this construction has nonspherical universal cover. This verifies the explicit sphere-cover exclusion required by the canonical cap-bundle supplier before that supplier is applied.

The canonical based Jordan caps form a Hausdorff proper disk bundle on positive bands by [F2], and lifted one-field transport gives a coherent regular cap family; the exact finite-clock derivative bound forces an infinite centre track, adjustment of recurrence gives one central plaque, and the no-hit and common-plaque arguments yield nested source disks and paired quotients. A closed positive transversal can avoid the whole lateral fence while crossing the inward leafwise annulus, and its compact oriented one-manifold pullback then gives a one-sign boundary contradiction by [F2]; hence the reduced leaves have no closed transversals, and by [F3] they are compact.

Open transversal saturation transfers the absence of closed transversals to the original leaf $L$, and the intrinsic-noncompact-to-transversal clause of [F3] makes the original $L$ compact; the finite-generator compact-nearby-leaf graph argument of [F3] prevents any distinct compact reduction leaf at positive parameters, preserving the identification with the original $L$; the recurrent $\mathrm{\Pi }$-side leaf $B$ is noncompact and has a strict positive closed return, so its open mutual-accessibility class $S$ is distinct from $L$, contains basepoints tending to $L$, and saturation of the closure gives $L\subseteq {\mathrm{\partial }}_{M}S$.

Therefore a leaf with a nonzero limitwise-nullhomotopy class on one side and base point is compact and lies in the ambient topological boundary of a distinct mutual-accessibility component $S$ with $S\cap L=\mathrm{\varnothing }$. The explicitly supplied finite surface, generic-loop, spherical-stability and transversal lemmas discharge every prerequisite used, the sphere-cover exclusion was verified in step 2.1 before cap development, and only the standing countable choice from [F5] is consumed.