A primitive pi torus collar has contracting longitude and exhausting plane caps

The meridian $\beta$ has identity one-sided holonomy by [F1]. Suppose $H(t)=t$ at a sufficiently small positive $t$. Both generators $\beta ,\lambda$ then fix $t$. Construct the nearby graph directly: cut $L$ along these generators, continue the plaque through $t$ along a finite tree of paths to a finite chart cover of the cut polygon, and shrink the base interval once so the finitely many transports and overlap homotopies are defined. Each overlap difference is a word in the two generators; holonomy invariance identifies its transverse transport with that word, which fixes $t$. Thus the local plaque sections agree, including across the polygon edges. They give a compact $C^2$ graph over $L$, contained in the leaf through $t$. Its inclusion is locally open in that intrinsic leaf and its image is intrinsically compact, hence closed; connectedness makes the graph the whole leaf. Collar projection would make its transported meridian essential, contradicting the nullness supplied by [F1]. This existence argument is separate from [F2]'s assertion about leaves already known compact. Hence $H$ has no positive fixed point on one small connected interval. Replacing $\lambda$ by its inverse if necessary gives $0<H(t)<t$; its iterates decrease to zero, since any positive limit would be a fixed point.

Finite plaque transport over a cut fundamental polygon of the torus gives the actual collar suspension: $\beta$ identifies the meridian edges without transverse change and $\lambda$ identifies the longitude edges by $H$; the construction uses finitely many compact chart relations, all valid after one common shrink. For each $t$ the longitude circuit thickened by the meridian coordinate gives an embedded leafwise annulus $A_t$ from ${\beta }_t$ to ${\beta }_{H(t)}$, whose base projection travels once over the complementary torus annulus. Two disjoint embedded null circles in a nonspherical oriented leaf joined by an annulus have nested Jordan disks whose difference is that annulus: otherwise the two disks and the annulus would form an open-and-closed sphere leaf.

The nesting direction is locally constant in $t$ by compact cap transport and the disjointness of the boundary circles. It cannot be shrinking: if the disk of ${\beta }_{H(t)}$ lay inside the disk of ${\beta }_t$, iteration would place every ${\beta }_{H^n(t)}$ inside the compact disk of ${\beta }_t$, but these circles approach the original torus, which is disjoint from that disk, contradicting the positive ambient distance between the two compact sets. Hence the disk relation is $D_{H(t)}=D_t\cup A_t$ for every small $t$.

The union of the increasing disks $D_{H^n(t)}$ is the entire nearby leaf: given any point of that leaf, join it to a point of $D_t$ by a compact intrinsic path; its ambient image is compact and disjoint from the original compact torus, hence has positive distance from it, while the late boundary circles ${\beta }_{H^n(t)}$ lie arbitrarily close to the torus and avoid that path, so the path cannot leave the late disk and its endpoint belongs to the union. An increasing union of disks with each compactly inside the next is a plane: choose successive disk and annulus homeomorphisms to concentric disks of radii $n$ and glue them, the exact differences $A_{H^n(t)}$ providing the annuli. These annuli lie in collar heights tending uniformly to zero, so they have no limit points away from the original torus, while every point of the torus is approached because their meridian and longitude base projections cover the whole torus; the compact initial disk contributes no intrinsic end-limit points. Thus the leaf limit set is exactly the original torus. The limit-set convention is independent of the chosen exhaustion: any intrinsic compact subset is contained in a sufficiently late disk because the disks form an increasing open cover and compactness selects finitely many whose maximum contains it; intersecting closed exhaustion tails and the escaping-sequence definition agree in the compact metric ambient space by choosing one point from each shrinking rational neighbourhood and tail, using the standing countable choice of [F3].