A first saddle lobe admits a collar-fixed center-saddle cancellation

Choose one smooth positively transverse field $V$ near the compact image of $\mathrm{\Gamma }$. In each foliation box fix the intrinsic plaque of $L$ through its arc of $f(\mathrm{\Gamma })$, with transverse coordinate ${\zeta }_i=0$. The equation ${\zeta }_i({\mathrm{\Phi }}_{-s}f(x))=0$ has a unique short $C^2$ root, since its $s$ derivative is nonzero. Its projected point is $B_C(x)$, and the assigned plaque and common short $V$ orbit make the roots agree on overlaps. The specified smoothing collar and null filling make the frontier leafwise null, so its identity holonomy returns the same assigned plaque after a circuit; this constructs $B_C$ on a thin neighbourhood $C$ of all of $\mathrm{\Gamma }$, including $q$. Choose a regular inner circle inside $C\cap \mathrm{\Omega }$. The given rounding collar and cap supply a continuous filling of its projection in the intrinsic leaf. Prescribe $B_C$ on its whole boundary collar, apply the relative cap approximation in [F4] to preserve that germ on a smaller open collar, and extend by $B_C$ outside this inner circle to form an enlarged $C^2$ cap $B:\widehat{W}\to L$. Thus $B=B_C$ on a neighbourhood of $\mathrm{\Gamma }$ and the eventual source boundary; no saddle jets are flattened in making this extension. Apply [F3] with the same $V$ on this fixed cap before choosing the final block. It gives a jointly $C^2$ product $P:\widehat{W}\times J\to M$, and its actual section $T=t_C$ satisfies $f(x)=P(x,T(x))$ pointwise on $C$. Since $T=0$ on compact $\mathrm{\Gamma }$, shrink $C$ so its compact section ranges lie in $J$.

Near the center choose the pullback $H$ of a genuine transverse foliation coordinate. If $\omega$ is a local nonvanishing defining one-form, then $f^{\ast }\omega =\alpha dH$ with $\alpha \mathrm{\ne }0$. At a characteristic zero its derivative is $\alpha \mathrm{Hess}H$, so the nondegenerate-center hypothesis makes $H$ a genuine $C^2$ Morse minimum after choosing the increasing orientation. The same argument makes the actual section $T$ Morse at $q$. On each regular full-circle annulus, $H$, $T$ and $u_0$ factor through the regular quotient coordinate, with positive one-variable derivatives after fixing their common coorientation. Use $H$ only near $p$, $T$ near the outer annulus and $q$, and interpolate their positive derivatives on a compact middle quotient interval. The positive affine amplitude and offset of $H$ are free: choose its endpoint value below the fixed upper value and its amplitude sufficiently small, then choose a positive interpolating derivative with the required integral. Integrating this derivative joins the two primitives with matching germs. This gives a $C^2$ primitive $T_0$, equal to actual $T$ near $\mathrm{\Gamma }$ and its exterior collar, with just the minimum and saddle. It uses $u_0$ only on regular annuli and assumes no nondegenerate Hessian for $u_0$ at $p$.

Raise the minimum of $T_0$ toward $0$ by a strictly increasing reparameterization on the inner circle component, equal to the identity on a neighbourhood of the frontier value $0$; equivalently choose the small inner amplitude and positive interpolation in step 2.1 to make its minimum as close to $0$ from below as needed. This preserves the regular circle leaves and the actual $T$ germ near $\mathrm{\Gamma }$ and leaves the exterior negative branch unchanged. Choose $a$ in that exterior branch tube and $b>0$, with $a<T_0(p)<0<b$, so all these boundary levels and their buffers lie in $C$. Smooth $T_0$ in $C^2$ on a compact neighbourhood in the ambient source plane, without imposing boundary equality, to a smooth $f_0$. For sufficiently small error it has precisely one minimum and one saddle near $p,q$ and no other critical points: outside small critical balls the gradient has a positive minimum, while in each ball its Hessian remains within half the least singular value of the original Hessian. The map $x\mapsto x-H_{\ast }^{-1}\mathrm{\nabla }{f}_0(x)$ is then a contraction on a smaller ball, with its displacement of the old zero smaller than the inward margin, so its iterates converge to the unique new zero; the Hessian inertia is unchanged. Keep $a<f_0(p)<f_0(q)<b$ and $\mathrm{\mid }T-f_0\mathrm{\mid }<(b-a)\mathrm{/}3$ on the eventual boundary collar.

Build an adapted descending field for $f_0$ in small Morse charts and regular connecting tubes using [F2]. The original inward saddle branch crosses a finite regular tube into a compact center sublevel disk, and the outward branch crosses a finite tube through the prescribed transverse exit. Strict derivative signs on these compact tubes persist under the chosen $C^2$ approximation. Route the corresponding local saddle rays through these two tubes, patching positive scalar derivatives in regular charts; the inward ray enters the center disk, whose decreasing flow has only the minimum as possible limit, and the other ray exits below $a$. Thus exactly one of the saddle's two descending branches limits on the minimum. Reverse sign to obtain a smooth upward adapted field $Y$; on the boundary buffer choose its regular pieces sufficiently close to $\mathrm{\nabla }{T}_0$ that $dT(Y)>0$, since $T_0=T$ there. These are strict signs on compact regular regions; no equality of perturbed orbits or derivative estimate for a cancelled scalar is assumed.

Construct the actual interval-face block inside $\widehat{W}$. At $f_0=a$ take a short exterior incoming interval straddling the outward descending branch. Its two endpoints, chosen on opposite sides of that branch, have upward trajectories avoiding the saddle and passing through its two outgoing arms. Follow these trajectories to $f_0=b$, and join them by the outgoing level interval that runs around the near-frontier circle component. The enclosed region is the incoming interval strip with the center zero handle and saddle one handle added: below the center there is the exterior interval strip, the center adds one disk component, and the saddle band attaches once to that disk and once to the strip. After this joining the outgoing face is one interval and the region is a disk, with precisely $p,q$ in its interior and two trajectory sides. This also describes its boundary without assuming fictitious circle faces. The regular frontier arcs, the saddle chart and the chosen exit tube form a finite compact collection. Choose $a,b$, side widths and the approximation error sufficiently small that the whole boundary buffer remains in $C$ and is regular, disjoint from $p,q$ and the closed connecting orbit. Thus the actual section $c=T$ is defined on the entire boundary collar, $dc(Y)>0$ there, and its face errors from $f_0=a,b$ are less than $(b-a)\mathrm{/}3$.

Apply [F1] to this actual disk triad, its smooth $f_0,Y$ and its $C^2$ scalar $c=T$ on the full boundary collar. The unique orbit in step 4.1 is the sole attaching-belt intersection. The supplier glues an abstract regular rectangle to the two sides only for the two-circle-face Milnor cancellation theorem; the field modification is supported inside the actual $W$, so unchanged tangent sides prevent a modified trajectory from leaving $W$. Its smooth product coordinates then integrate a positive density with variable face values $A(y),B(y)$ and with a transverse side blend. Integration by parts gives a $C^2$ primitive while preserving the entire boundary germ. Consequently there is $v\in C^2(W,\mathbb{R})$ with $dv\mathrm{\ne }0$ and $v=T$ on an open collar of all of $\mathrm{\partial }W$. Round corners within this regular collar, retaining the scalar germ and the enclosed lobe and saddle.

Choose an open interval $I=(\mathrm{\ell },r)$ with the compact boundary section range $S⋐I⋐J$, and choose $c<d$ inside $I$ with $S\subset (c,d)$. A positive smooth function $h$ can equal $1$ near $[c,d]$ and have tail integrals ${\int }_{-\mathrm{\infty }}^{c}h<c-\mathrm{\ell }$ and ${\int }_d^{\mathrm{\infty }}h<r-d$: join $1$ to positive exponential tails on arbitrarily short intervals and make the remaining tail integrals arbitrarily small. Set $\vartheta (s)=c+{\int }_c^{s}h(t)dt$. Then ${\vartheta }^{\mathrm{\prime }}=h>0$, $\vartheta$ is the identity on $[c,d]$, and $\vartheta (\mathbb{R})\subset I$. Hence $w=\vartheta \circ v$ is $C^2$, $dw=h(v)dv\mathrm{\ne }0$, and $w=T$ on the exact boundary collar. This controls the whole cancelled scalar range without claiming it is close to the old scalar.

Define $f_{\mathrm{n}\mathrm{e}\mathrm{w}}(x)=P(x,w(x))$ on $W$ and extend by $f$ on its open boundary collar and outside $W$. On the collar, steps 1.1 and 7.1 give $P(x,w(x))=P(x,T(x))=f(x)$, so this is a $C^2$ map with the same outer boundary map. Since $d{P}^{-1}(TF)=\ker (dt)$, the characteristic distribution of its graph is $\ker (dw)$, which is regular everywhere in $W$. Exactly the original one center and one saddle have been removed there; every singularity outside $W$ is unchanged. No interior homotopy is required. Only the standing choice in [F5] is inherited through the suppliers, and all extra collars, rectangles, cutoffs and tail parameters are finite choices.