A finite characteristic circuit has C² regular port traces

The circuit data are finite: by [F2] the frontier consists of finitely many saddle points and finitely many compact regular edges, and its directed polycycles form a finite cover of the edge set. Each regular edge $E$ is a nonconstant trajectory of the characteristic field, so the first integral $u$ of [F1] is constant along $E$ and $du\mathrm{\ne }0$ on $E$. Near a regular point the level sets of $u$ are the characteristic leaves, and the port sections chosen in the statement are transverse to the characteristic foliation there.

Port parameterization. Fix a port section $\mathrm{\Sigma }$ through a regular point $p$ of an edge. In a flat chart containing $\mathrm{\Sigma }$ the scalar $g(\xi ,t):=u(\xi )-t$ is $C^2$, $g(p,u(p))=0$ and ${\mathrm{\partial }}_{\xi }g\mathrm{\ne }0$ along $\mathrm{\Sigma }$ because $\mathrm{\Sigma }$ crosses the level set transversally; by [F3] the level $t$ meets $\mathrm{\Sigma }$ in a unique point depending jointly $C^2$ on $t$ near $u(p)$. Taking $t=0$ at the frontier level shows that the ports, including the frontier port, depend $C^2$ on the level. On overlaps of two flat charts the two first integrals differ by the $C^2$ transverse transition of $F$, so the parameterization is chart-independent and the transversality of each port to the ambient foliation is preserved.

On each compact trimmed regular edge the C² first integral is a submersion. A finite chain of its inverse-coordinate rectangles supplies local level strips. To obtain exact overlaps, choose a reference C² parametrization of the edge; neighboring strip candidates agree on it at level zero, and in their common plaque coordinate blend them with a fixed source cutoff on an overlap, equal to the corresponding candidate on its end collars. The reference tangent has one strict sign, so after a common shrink the blended tangent keeps that sign. Its transverse label is kept fixed throughout. Finite such blends give a jointly C² regular strip, including its endpoint collars at the ports. This constructs compatible pieces before applying [F4].

At a saddle choose one target foliation box containing the images of the two sufficiently close ports and of the intervening saddle passage. For each nearby source level the passage lies in a single target plaque; matching the two port transverse coordinates in this target box therefore gives a C² local transverse transition. This concerns the ambient plaque label and the regular endpoint collars of step 3.1, and supplies no regular source strip across the saddle.

The nearby closed loops. By [F2] the chosen period annulus carries its prescribed closed characteristic loops with a $C^2$ transverse trace; for every level $t$ in some one-sided interval $(0,\epsilon )$ the corresponding loop follows the finite itinerary and closes up. The composite transverse return map of the itinerary is obtained by composing the finitely many port and strip transitions of steps 2.1–4.1 around the itinerary; it is a $C^2$ germ of a real function, and each closed level loop returns to its own level, so the return map fixes every $t\in (0,\epsilon )$.

Fixed on an interval. A $C^2$ function that fixes every point of a nondegenerate interval equals the identity on that interval; hence the one-sided composite transverse return map is the identity on $(0,\epsilon )$, which is (c).

All constructions selected finitely many charts, edges, ports and intervals; the root and transport theorems used are choice-free, so nothing beyond the standing hypothesis [F5] is invoked, and (a), (b), (c) follow from steps 2.1, 4.1 and 6.1.