A null-transversal disk has a minimal one-sided cycle

By [F1] the disk map is in relative generic position and its finitely many characteristic singular points have distinct ambient leaf images; applying the minimum-selection supplier [F2] to this separated disk yields a source characteristic cycle $P$ with nonidentity full holonomy whose bounded simple-cycle domain is inclusion-minimal among all source characteristic cycles with nonidentity holonomy. The cycle $P$ is either a regular closed characteristic orbit or a finite saddle polycycle, and no disk replacement is used in its selection.

Suppose $P$ is regular or its bounded homoclinic side occupies one saddle sector, and its inward germ is nonidentity. On that side, the finite regular strips and, in the homoclinic case, the single saddle-sector passage give a genuine one-circuit source return map $R$. Finite target plaque transport identifies it with the inward ambient holonomy, up to conjugacy and possible inversion. Choose an arbitrarily small inward parameter $r$ with $R(r)\mathrm{\ne }r$; reverse the characteristic direction if necessary so $R(r)>r$. Over one return strip, use regular $C^2$ first-integral strip coordinates $(s,u)$ with $s$ increasing along trajectories and $u$ constant, and choose a strictly decreasing graph $u=f(s)$ from $r$ to $R^{-1}(r)$. Its endpoints match after the return identification. Choose matching endpoint derivatives and smooth the seams while retaining $f^{\mathrm{\prime }}<0$. Its image is a simple $C^2$ source circle $C$ inside the bounded side of $P$, following that one-sector itinerary once. The field crosses $C$ toward the inward side $u>f(s)$; the bounded Jordan disk $K_C$ is therefore positively invariant and lies strictly inside the domain of $P$. The construction is at positive regular parameters and does not smooth through the saddle itself.

If the bounded side of $P$ occupies three saddle quadrants, use the area-minimizing conclusion of the selection supplier [F2] and apply [F3] to this same cycle in the unchanged disk. It supplies identity inward holonomy directly, as well as the inner one-quadrant loop $Q$ with identity full holonomy and the equality of the realized inward return with the inward $P$-holonomy. Since $P$ has nonidentity full holonomy, its opposite-side germ is nonidentity by [F4].

In the regular or one-sector case of step 2.1, restrict the original disk map to $K_C$, using a disk parametrization of this regular planar Jordan domain. Its characteristic singularities are the original finitely many nondegenerate interior singularities; their ambient leaves remain distinct, and its boundary $C$ is everywhere transverse to the characteristic field, hence its image is a closed transversal to $F$. Thus this restricted generic disk satisfies the hypotheses of [F2]. That supplier gives a simple characteristic cycle $Q$ with nonidentity full holonomy and bounded domain inside $K_C$. This is a cycle in the original source disk, strictly inside the domain of $P$, contradicting its original inclusion-minimality. Nonidentity of $Q$ comes from the nonidentity-cycle supplier, not from the center-period-annulus frontier alternative. Consequently the inward germ of $P$ is identity, and its full nonidentity germ is nonidentity on the opposite side.

Therefore in every case the selected cycle $P$ has identity holonomy on the inward half-transversal and nonidentity holonomy on the opposite side, and the selection is inclusion-minimal among the source characteristic cycles with nonidentity holonomy and bounded simple-cycle domains. The polycycle endpoint additionally uses the separate rounding construction of [F3] once its family hypotheses are supplied, and a vanishing-cycle application requires in addition a proved inward family and leafwise caps; interior singularities are retained and no claim that all interior trajectories are closed is made. The selection and the case analysis use only finitely many source cycles, ports and sections together with the cited suppliers, hence only the standing countable choice from [F5].