A null simple center frontier supplies the exact cancellation scalar

The source interior of the frontier occupies exactly one saddle quadrant of $q$: the two coordinate axes of the saddle normal form of [F2] give four quadrants, and if the source interior occupied three of them, the two unused characteristic separatrix half-rays would lie inside $\mathrm{\Omega }$, whereas every point of $\mathrm{\Omega }$ other than the center $p$ lies on a periodic orbit, and a separatrix half-ray is not a periodic orbit. Hence exactly one quadrant is occupied, and the maximality of the nested-circle basin and the absence of other zeros make $p$ the only interior center.

The rounded image of $\mathrm{\Gamma }$ being null in its leaf makes the holonomy germ of $\mathrm{\Gamma }$ the identity on both sides: the leafwise class of $\mathrm{\Gamma }$ is trivial, so by [F1] the holonomy of $\mathrm{\Gamma }$ is the holonomy of a null loop, namely the identity germ, and the same holds for the opposite side.

Choose one smooth positively transverse ambient field $V$ near the compact image of $\mathrm{\Gamma }$, by finitely many positive chart fields and smooth nonnegative weights. In each foliation box let ${\zeta }_i=0$ be the assigned plaque of the intrinsic leaf $L$ through the corresponding arc of $f(\mathrm{\Gamma })$. The equation ${\zeta }_i({\mathrm{\Phi }}_{-{\sigma }_i(x)}f(x))=0$ has a unique short $C^2$ root because its derivative in ${\sigma }_i$ is nonzero. On overlaps the same $V$ orbit and assigned intrinsic plaque give the same root and projected point. The two-sided identity holonomy of step 1.2 returns the same plaque branch after one circuit, so these expressions give a $C^2$ projection $B_C$ on a sufficiently thin neighborhood $N$ of $\mathrm{\Gamma }$, including its saddle, with $B_C=f$ on $\mathrm{\Gamma }$. No embeddedness of the whole leaf or exclusion of distant branches is used. Choose a regular inner circle $C$ of the basin inside $N$ and an enlarged source disk $\widehat{W}$ with exterior boundary inside $N$. The projection annulus, prescribed leafwise rounding collar and given null filling supply a continuous leafwise filling of $B_C{\mathrm{\mid }}_C$. Attach the prescribed $B_C$ germ on a collar of $C$, apply [F6] to the remaining filling disk, and extend by $B_C$ on $\widehat{W}$ outside $C$. This gives one $C^2$ leafwise cap $B$ agreeing with the entire projection germ near $\mathrm{\Gamma }$ and the exterior boundary; no flattening of the saddle jets is performed.

Apply the fixed-cap transverse-product construction of [F3] to $B$ with this same field $V$. The actual trace $f$ on $N$ is obtained along its short $V$ orbits from $B_C=B$, so its transported section $T=t_C$ satisfies $f(x)=P(x,T(x))$ exactly there. The product interval is fixed first; since $T=0$ on compact $\mathrm{\Gamma }$, shrink $N$ so its section range is compactly inside that interval. Its differential annihilates the characteristic line field. Near $q$ a genuine foliation one-form pulls back as $c(x)dT$ with $c(q)\mathrm{\ne }0$, so differentiation at the zero gives a nonzero scalar multiple of the Hessian of $T$. Characteristic nondegeneracy makes this Hessian indefinite and invertible, and $T(q)=0$. This constructs the actual cap section directly.

Choose the transverse sign so $T<0$ in the basin quadrant. Near the center choose the pullback $H$ of a genuine foliation transverse coordinate, with sign making its Hessian positive definite; the same nonzero-factor calculation as in step 3.1 makes $H$ a $C^2$ Morse minimum. On the regular nested-circle annulus [F3] gives a $C^2$ quotient coordinate with full connected circle fibers. Both $T$ on an outer annulus and $H$ on an inner annulus factor through it with positive derivative. Scale and translate $H$ by a positive affine change so its value difference to the prescribed outer coordinate exceeds the two fixed endpoint-collar integrals. On the intervening compact quotient interval choose a positive $C^1$ derivative matching the endpoint derivatives; after shortening the endpoint collars, a positive middle bump obtains the exact required integral. Integration gives a $C^2$ scalar equal to $H$ near $p$ and to the actual $T$ on $N$; no differentiability of a quotient coordinate at the critical center value is assumed. Call it $u$. It is a first integral on a full neighborhood of the closed lobe, has only the minimum $p$ and saddle $q$, and agrees with the actual cap section on the whole exterior collar.

For the standard Euclidean metric, the negative gradient $-\mathrm{\nabla }u$ has two saddle unstable half-rays by the local hyperbolic-gradient picture of [F3]. Exactly one of them lies in the basin quadrant, and the other lies in the opposite sign-negative quadrant outside $\mathrm{\Omega }$. The inside ray cannot leave $\mathrm{\Omega }$: $u$ strictly decreases along it while the whole frontier $\mathrm{\Gamma }$ has value zero, so starting from a negative level it remains in a compact inner disk. Were its limiting value greater than $u(p)$, a compact regular level band would satisfy $\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }\ge m>0$, contradicting $du\mathrm{/}d\tau =-\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2\le -m^2$ along the ray; hence its value tends to $u(p)$, any accumulation point has value $u(p)$, and the only such point is $p$, so the ray tends to the center. The other ray is regular immediately after leaving a small saddle chart and crosses a short transverse exit section before reaching any other singularity.

The fixed cap $B$, product $P$ and section on the exterior collar were built with the same field $V$ in steps 2.1 and 3.1. The scalar $u$ was extended inward while retaining that section exactly, rather than reparametrized across possibly disconnected level components. Thus its cap-product collar identity is pointwise and the branch conclusions of step 5.1 apply to this actual $u$. The cap is a map into the intrinsic leaf and may have self-intersections; its finite compact image and prescribed collar germ suffice.

Therefore a maximal nested-circle basin with a single simple homoclinic frontier and a null leafwise rounded image supplies: a $C^2$ first integral on a full neighbourhood of the closed lobe and saddle that agrees with the transported fixed cap section on a regular exterior collar, and a Euclidean negative gradient with exactly one unstable half-trajectory entering the lobe and tending to the center while the other exits through a regular local section. These are precisely the scalar, branch, collar and fixed-cap hypotheses of the conditional simple-lobe cancellation carrier, and the construction used only finitely many boxes, collars and bump parameters together with the two sibling suppliers, hence only the standing countable choice from [F5].