A characteristic disk with essential boundary data produces a vanishing cycle

Preserve the boundary and separate the zeros. Fix a closed zero-free outer collar strictly inside the given regular collar. In case (a) the characteristic field is transverse to its boundary circle; in case (b) it is tangent and nonzero there, so that circle is a regular characteristic orbit even when its ambient image is not immersed. Its ambient leafwise class is the prescribed nonzero class. Apply the separation supplier of [F2] relative to this fixed collar. The result has the same finite singular set, Hessian types and outer map, and its singular images lie in distinct ambient leaves.

Quadratic normalization with two derivatives. At each singular point $p$ choose a disjoint interior source ball mapped into one foliation box, write $h=(Y,u)$ in that box and $w=x-p$, and set $Q(w)=u(p)+\frac{1}{2}{w}^{T}Hw$, where $H=D^{2}u(p)$ is invertible. Put $R(w)=u(p+w)-Q(w)$. Uniform continuity of $D^{2}u$ and two applications of the segment integral form of the mean-value theorem give, for $\mathrm{\mid }w\mathrm{\mid }\le \epsilon$, $\mathrm{\mid }R(w)\mathrm{\mid }\le \delta (\epsilon )\mathrm{\mid }w{\mathrm{\mid }}^2$, $\mathrm{\mid }DR(w)\mathrm{\mid }\le \delta (\epsilon )\mathrm{\mid }w\mathrm{\mid }$, and $\mathrm{\parallel }{D}^{2}R(w)\mathrm{\parallel }\le \delta (\epsilon )$, with $\delta (\epsilon )\to 0$. Choose a fixed smooth radial cutoff ${\chi }_{\epsilon }$ equal to zero for $\mathrm{\mid }w\mathrm{\mid }\le \epsilon \mathrm{/}3$ and one for $\mathrm{\mid }w\mathrm{\mid }\ge 2\epsilon \mathrm{/}3$, with $\mathrm{\parallel }{D}^j{\chi }_{\epsilon }\mathrm{\parallel }\le C_j{\epsilon }^{-j}$ for $j=1,2$. Replace $u$ by $\tilde{u}=Q+{\chi }_{\epsilon }R$, retaining $Y$. On the transition annulus the displayed estimates and the product rule imply $\mathrm{\mid }D({\chi }_{\epsilon }R)\mathrm{\mid }\le C\delta (\epsilon )\mathrm{\mid }w\mathrm{\mid }$; elsewhere this bound follows directly. Since $\mathrm{\mid }Hw\mathrm{\mid }\ge a\mathrm{\mid }w\mathrm{\mid }$ for some $a>0$, choose $C\delta (\epsilon )<a\mathrm{/}2$. Thus $D\tilde{u}\mathrm{\ne }0$ for $w\mathrm{\ne }0$, and its only critical point is $p$, with the same value and Hessian. The changes in derivatives of orders zero, one and two are bounded respectively by $C\delta (\epsilon ){\epsilon }^2$, $C\delta (\epsilon )\epsilon$, and $C\delta (\epsilon )$, so the modification is arbitrarily $C^2$ small. The compact chart-image margin and continuity of composition with the fixed $C^2$ inverse chart therefore make it a genuine $C^2$ disk map, unchanged outside the ball. The same gradient estimate holds during interpolation from $u$ to $\tilde{u}$. Performing these finitely many modifications fixes every singular image, its type, and the outer collar.

The regularity gained by normalization. A constant linear change diagonalizes each $H$. Near a saddle the zero level of its exact quadratic germ consists of two straight lines with four distinct rays; near a center its levels are ellipses. Every regular separatrix segment away from the saddle is a $C^2$ regular level arc by [F6]. A finite homoclinic edge therefore extends to its saddle endpoints as piecewise-$C^2$ arcs with distinct tangent rays, in the actual smooth source coordinates. Its composition with the disk map is piecewise $C^2$, although the ambient image need not be immersed. This is a property of the modified disk, not a regularity assertion about the original $C^2$ saddle germs. For example $xy+\mathrm{\mid }x{\mathrm{\mid }}^{2+\alpha }$, $0<\alpha <1$, need not have $C^2$ zero-level branches; no $C^2$ Morse-coordinate change has been assumed.

Verify the source-frontier alternatives. Select a center, which exists by $c-s=1$, and use [F3] for its maximal nested-circle annulus. Every connected saddle frontier maps into one ambient leaf: each regular edge lies in a leaf and its continuous endpoint lies in the same intrinsic plaque branch. Distinct singular ambient leaves thus force this graph to have one saddle vertex. There are only two stable and two unstable rays at that vertex; uniqueness of regular trajectories makes its graph consist of one or two simple homoclinic edges, each using one ray of each type. The corresponding loops are either side by side or nested. The swept region $\mathrm{\Omega }$ is the increasing union of the bounded periodic disks. It is an entire bounded component of the complement of the frontier: it is open and connected there, and any relative boundary would belong to $\mathrm{\partial }\mathrm{\Omega }$, which is exactly the frontier. For two side-by-side loops it is one lobe, and its frontier is that simple circuit. For two nested loops it is the region between them; the inner bounded disk $K$ excludes the selected center. Its interior at the saddle occupies one quadrant: if it occupied three, the other two separatrix rays, and hence the outer loop, would lie inside it by uniqueness, contradicting the nesting. The one-quadrant count and strict inner search of [F3] apply to this genuinely piecewise-$C^2$ circuit. They give a center in $K$ and strictly fewer interior saddles whenever another nested frontier is selected. A frontier that reaches the old invariant boundary is that original simple circuit. Consequently finite inner descent yields a simple saddle circuit or a regular orbit as endpoint.

Verify transverse traces and essential endpoints. Near the chosen center the small characteristic loops are null in one plaque. If an interior regular annulus loop is essential, restrict the $C^2$ annulus product trace to the compact interval between one near-center circle and that loop; [F4] gives a vanishing cycle. Otherwise every circle of the annulus is null. For a simple saddle endpoint, [F4] supplies a jointly $C^2$ family extending to a leafwise rounded circuit and preserving the leafwise classes of the earlier circles. If that endpoint is essential, take one nearby null circle and its endpoint in this family and apply the first-essential-loop lemma. For a regular endpoint, use a finite chain of regular $C^2$ level rectangles and transverse return roots from [F6]; their return map is the identity on the annulus side because all prescribed circles close. It gives the same jointly $C^2$ trace down to the regular endpoint. If ambient edge images are not immersed, finite plaque-coordinate chord replacements and corner roundings from the construction in [F4] regularize them, preserving transverse labels and leafwise classes. The first-essential-loop argument again applies to an essential endpoint, including the prescribed essential leafwise outer boundary.

Exclude the other regular endpoints. If a regular endpoint is leafwise null, [F6] gives identity holonomy on both sides. A short source transversal has nonzero pulled-back transverse derivative, and its return germ is conjugate, by that transverse coordinate, to the ambient holonomy germ along the endpoint. It is therefore the identity on a full two-sided interval. The regular level rectangles close into a band of circles across the endpoint, contradicting maximality. The original outer transverse boundary cannot be an endpoint: in its compact collar the characteristic radial component has one fixed sign and is bounded away from zero, whereas a periodic curve entering the collar has a radial minimum at which that component vanishes. The complete frontier classification in [F3] excludes other alternatives. These arguments also apply in an invariant inner search disk: its piecewise characteristic boundary cannot be crossed, and reaching it gives the already selected simple circuit rather than an additional regular alternative.

Verify every cancellation datum for a null simple endpoint. The remaining endpoint is a leafwise-null rounded simple homoclinic circuit. Its swept bounded lobe has just the chosen center and no other characteristic zero: every other point of the lobe belongs to a full regular circle of the annulus, because the swept disks increase from the small center disk through the annulus product. More explicitly, the product between any two periodic circles is a compact embedded annulus; its interior is open and its image is closed in the connected region between their Jordan boundaries, so it fills that region. Taking the increasing union, together with the small center disk, fills the entire swept lobe. The lobe occupies one saddle quadrant; if three were occupied, the two unused nonperiodic separatrix half-rays would be inside this circle-foliated region. Its source frontier is embedded and piecewise $C^2$ by step 3.1. Fix the leafwise rounding and one null filling. The scalar supplier [F5] constructs a $C^2$ leafwise cap equal to the actual projection germ on a neighborhood of the frontier, including the saddle, using the two-sided identity holonomy. Its transverse product gives a section equal pointwise to the original disk map on that whole neighborhood. Interpolation of positive derivatives on the regular circle quotient joins this actual section to a genuine center first integral, giving a $C^2$ first integral on the full closed lobe and its exterior collar. With the center sign chosen as a minimum, the inward unstable half-ray of its Euclidean negative gradient stays in a compact inner sublevel disk and tends to the sole center: on any compact regular level band $\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }$ has a positive minimum and $du\mathrm{/}dt=-\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2$ excludes any other limiting value. The opposite half-ray lies outside the lobe and meets a short regular exit section. Thus the cap, exact collar identity, full first integral, two gradient branches, isolated center and saddle, and piecewise-$C^2$ embedded source frontier required by the cancellation supplier are all supplied; none is inferred from genericity alone. Apply that supplier to remove precisely one center and one saddle inside a compact support block while retaining the map on an open collar of its boundary and outside it.

Renew the contract before the next search. The replacement is a $C^2$ map with the original outer collar and boundary data, exactly the unchanged zeros outside the cancellation block, and no zero inside it. Re-separate those finitely many singular images relative to the original outer collar and repeat the quadratic normalization of steps 2.1–3.1. Separation preserves the critical germs up to constants, and normalization fixes the resulting critical images and Hessians; neither operation creates a zero. Hence the total saddle count has decreased exactly by one. Start a new maximal-annulus search on this normalized disk; the topology, trace and exact cancellation verifications in steps 4.1–7.1 apply anew. No old center basin, separatrix connection or inner search disk is asserted to survive these perturbations. This establishes the exact maximal-center-frontier contract used in [F1] at every iteration: the complete frontier alternatives, strict inner-disk descent, jointly $C^2$ essential-endpoint traces, and all conditional simple-lobe cancellation data with unchanged outer collar and exact count decrease.

At fixed total saddle count $S$, each nested inner search lowers its finite strict-interior saddle count, so it terminates in an endpoint dealt with in steps 5.1–7.1. Each cancellation lowers $S$, and step 8.1 renews every hypothesis before another search. This is the lexicographic finite rank in [F1]. At $S=0$ the full-disk count still gives a center; no saddle endpoint exists, and the regular or boundary alternatives of steps 5.1–6.1 must give a vanishing cycle. The original boundary remains throughout either the original closed transversal or the original essential leafwise loop. Thus the process terminates in a vanishing cycle in either case (a) or (b). Only the stated countable choice is inherited through the cited suppliers; all extra balls, cutoffs, normalizations and cancellations are finite choices.