The center-frontier selection and cancellation search has finite rank

Fix a smaller closed zero-free outer collar, so the separation supplier of [F1] applies with its required zero-free collar closure. Before searching, normalize each of the finitely many critical germs in disjoint interior balls. In a foliation box write the map as $(Y,u)$, let $w=x-p$, $H=D^{2}u(p)$ and $Q=u(p)+\frac{1}{2}{w}^{T}Hw$. For $R=u-Q$, continuity of $D^{2}u$ and the segment integral estimate give $\mathrm{\mid }R\mathrm{\mid }\le \delta (\epsilon )\mathrm{\mid }w{\mathrm{\mid }}^2$, $\mathrm{\mid }DR\mathrm{\mid }\le \delta (\epsilon )\mathrm{\mid }w\mathrm{\mid }$ and $\mathrm{\parallel }{D}^{2}R\mathrm{\parallel }\le \delta (\epsilon )$ on $\mathrm{\mid }w\mathrm{\mid }\le \epsilon$, with $\delta (\epsilon )\to 0$. Choose a 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 $Q+{\chi }_{\epsilon }R$, retaining $Y$. The product rule gives $\mathrm{\mid }D({\chi }_{\epsilon }R)\mathrm{\mid }\le C\delta (\epsilon )\mathrm{\mid }w\mathrm{\mid }$ and a $C^2$ change tending to zero. Since $\mathrm{\mid }Hw\mathrm{\mid }\ge a\mathrm{\mid }w\mathrm{\mid }$ for some $a>0$, choosing $C\delta <a\mathrm{/}2$ excludes every new zero, also during the interpolation; the value and Hessian at $p$ are unchanged. Smallness keeps the image in its foliation box. Thus all singular images remain separated and the outer collar stays fixed. A linear change diagonalizes $H$; the saddle critical level now has straight rays near the saddle. Elsewhere regular level arcs are $C^2$ by [F4], so every finite simple homoclinic circuit is genuinely piecewise $C^2$ in the source coordinates. Start the frontier search on this modified disk.

The period-frontier interface [F1] yields the frontier of the maximal center annulus. Distinct singular ambient leaves prevent a connected characteristic frontier from containing different saddle vertices; a one-saddle graph has at most two homoclinic edges; the basin is an increasing union of its bounded periodic disks, hence a whole bounded complementary component of that graph; a side-by-side two-loop graph has separate bounded components, so one center basin has only one lobe as its frontier; and a nested pair has a one-quadrant inner bounded disk $K$ containing a center. Searching from a center in $K$ regardless of the two image lobe classes, its trajectories cannot cross the invariant boundary, any further nested pair has its saddle strictly inside $K$ with inner disk excluding that saddle and hence strictly fewer interior saddles, and a later frontier reaching $\mathrm{\partial }K$ is that simple circuit. This is source topology and uses no inherited essential boundary class.

For a selected maximal annulus, near-center loops are null in a plaque. If any regular loop is essential, the first-essential-parameter construction produces a vanishing cycle using the C² period trace. Otherwise all regular loops are null. An essential simple saddle endpoint produces a vanishing cycle by the same first-essential construction; a null simple endpoint supplies one fixed cap and the cancellation of [F3]; a null regular endpoint has trivial two-sided ambient holonomy, so a regular source neighbourhood has a closed-orbit band across the endpoint, contradicting maximality; an essential regular endpoint again gives the first-essential trace. An outer transverse boundary cannot be the limit of periodic circles in its regular transverse collar, a regular leafwise boundary is the prescribed essential endpoint, and an isolated outer center is impossible in a disk: a small punctured centre neighbourhood is foliated by circles, so joining this cap to the original centre cap with the intervening product annulus would exhibit a compact boundaryless two-dimensional submanifold of the interior of the connected source disk, which is locally open in that disk and hence, by compactness, closed, a contradiction.

Define the rank as the pair $(S,s(K))$ with the lexicographic order, where $S$ is the total saddle count in the current full disk and $s(K)$ the number of interior saddles of the current invariant search disk. The inner-source-disk search of step 2.1 lowers $s(K)$ strictly; a null simple cancellation lowers $S$ exactly by one, leaves the original outer collar fixed and removes no other zero, after which the finitely many remaining singular images are re-separated relative to a smaller closed zero-free outer collar by [F1] and the quadratic normalization of step 1.1 is repeated before a new maximal-annulus search. Neither operation creates a zero; no old separatrix or basin is assumed to survive. At $S=0$ there is still a center by the index count $c-s=1$ of [F1], and the regular or boundary alternatives of step 3.1 yield an essential endpoint. Hence the full-disk theorem follows in finitely many cancellations, the iteration terminates in a vanishing cycle, and every regular annulus extension is absorbed into the one maximal family supplied by the frontier interface rather than an artificial new family.