Compatible arbitrary pi fence reduction

Start with the essential initial loop at $a_0=0$ and the closed/null family on $(0,b]$ represented by the full cyclic word $w_0$ of [F1]. If every positive level of the family has a simple lift to the universal cover of its leaf, terminate. Otherwise choose one positive parameter $r\le b$ at which the lift has a self-intersection; its actual lifted double point is one eligible marked pair. It splits the source word into two parameterized subpaths $c_t,d_t$, which are loops only at heights where the corresponding endpoints coincide; their genuine closed/null interval is selected in the next steps.

At $r$ both $c_r$ and $d_r$ are closed and null: each is the projection of a closed subpath of the closed lifted loop, and the universal cover is simply connected by [F3]. By finite compact-cap persistence in foliation boxes both remain closed and null on a neighbourhood of $r$, so the set of parameters near $r$ on which both are closed and null is a nonempty interval ending at $r$.

Let $(a,r]$ be the maximal common downward interval on which both $c$ and $d$ are closed and null relative to the current fence interval $[a_0,r]$. At $a$ both loops are closed by continuity. If $a>a_0$ and both were null at $a$, finite compact-cap persistence for both would extend the interval below $a$, contradicting maximality; if $a=a_0$ and both were null, then their product would make the current initial essential loop null, contradicting [F2] and the essentiality of the initial representative. Hence at least one factor is essential at $a$ by [F4]. Choose such a factor, retain only the family on $[a,r]$, and rebase and rescale it.

The new initial leaf need not be the old one, but the family stays on the same chosen transverse side and every positive displacement is genuinely closed and null; the rank of its cyclic word is strictly smaller by the finite-double-point representative of [F1]. Repeat the operation only when an actual positive-level lift is nonsimple; there are at most $N$ such operations, and at termination every positive lift is simple, because otherwise another rank-decreasing operation would be possible. Since the initial parameter belongs to the half-open interval, choosing $b$ arbitrarily small makes the reduced essential initial leaf arbitrarily close to $L$; no cut is projected below its common closed/null interval and no closedness of nullness is assumed.

At termination round the finitely many switched corners jointly before making regular cap charts: choose small target plaque boxes at the switch vertices, disjoint from every other retained zero-level arc except the two adjacent half-arcs, replace each corner by a $C^2$ regular joining arc, and transport the replacement by the prescribed transverse plaque label. The replacement is homotopic relative its two port collars and preserves nullness and essentiality; it creates no lifted collision because it is embedded in its small plaque box where no other retained arc lies; and it can be normalized back to one fixed fence by unique transverse roots and finite homotopy invariance of $\mathrm{\Pi }$, so all sufficiently small positive lifted boundaries remain simple and the boundary circles are genuinely $C^2$ rather than tacitly rounded corners. All operations are finite, hence only the standing countable choice from [F5] is used.