Self-indexing Morse functions exist

Start with the rearrangement theorem: by [F1] there are an adapted complete downward gradient-like field $X_0$ for $g_0$ and a Morse $g_0$ adjusted to $(f,X_0)$ whose critical levels, ordered by value, have nondecreasing indices; $X_0$ is also gradient-like for the initial $f$. List the critical levels of $g_0$ in increasing order of value as $E_1,\dots ,E_N$, with ${\mu }_1\le {\mu }_2\le \cdots \le {\mu }_N$ their indices. For a fixed index $k$, the levels with ${\mu }_i=k$ occur consecutively in this list, and the union of their points is again a set of critical points of common index $k$.

Merging two adjacent equal-index levels. Suppose the consecutive levels $E_i,E_{i+1}$ both consist of points of one index $k$, with values $c_i<c_{i+1}$, and no other critical value between them. Apply [F2] with $P:=E_i$, $Q:=E_{i+1}$ and a regular value $v\in (c_i,c_{i+1})$, which is legitimate since $\mathrm{ind}(p)=k\ge k=\mathrm{ind}(q)$ for all $p,q$: choose the field change arbitrarily small, as permitted by [F2], so that it also retains descent for the initial $f$ by the compact-band pairing estimate in the rearrangement proof. There is then a complete adapted field $X_1$ for both $g_0$ and $f$, equal to $X_0$ outside a prescribed neighbourhood of $g_0^{-1}(v)$, with no trajectory of $X_1$ having one limit in $E_i$ and the other in $E_{i+1}$. Then apply [F3] to the regular band $g_0^{-1}[a,b]$ spanned by regular values $a<c_i<c_{i+1}<b$ adjacent to the two levels, whose critical set in that band is exactly $E_i\cup E_{i+1}$, with equal prescribed values $a_P=a_Q\in (c_i,c_{i+1})$; this produces a Morse $g_1$ on the sub-triad with the same critical points and indices, both sets now at a common value, equal to $g_0$ outside a compact neighbourhood of the band and equal to $g_0$ plus a constant near each critical point, with $X_1$ still downward gradient-like. Extending by $g_0$ outside gives a function $g_1$ on $W$ with the same critical points and indices, equal to $g_0$ near $\mathrm{\partial }W$ and outside the band, for which $X_1$ is adapted and downward gradient-like, and in which the two levels have been merged into one.

Iteration. Repeat step 2.1: while two consecutive levels of the same index exist, separate them with the perturbation of [F2] and merge them with the exchange of [F3]. Each step strictly decreases the number of distinct critical levels and leaves unchanged the multiset of indices of the critical points; the number of levels is a nonnegative integer bounded below by the number of distinct indices present, so after finitely many steps all critical points of a given index $k$ lie at one common level $c_k$, and the levels $c_{k_1}<\cdots <c_{k_m}$ of the indices $k_1<\cdots <k_m$ that occur are strictly increasing with the index. Each modification is supported in a compact band around the two levels being merged or in a neighbourhood of one regular level, so the boundary behaviour near $\mathrm{\partial }W$ is unchanged throughout and every intermediate field is complete and adapted.

Reparametrization of the values. Let $g_m$ be the function obtained at the end of step 3.1, with critical points of index $k_j$ at the common level $c_{k_j}$, and let $X_m$ be its adapted field. Put $d_{k_j}:=(k_j+1)\mathrm{/}(n+2)$; these form a strictly increasing sequence in $(0,1)$. By [F4] there is a smooth increasing diffeomorphism $\psi :[0,1]\to [0,1]$ with $\psi =\mathrm{i}\mathrm{d}$ near $0$ and $1$, derivative one near every $c_{k_j}$, and $\psi (c_{k_j})=d_{k_j}$ for every $j$. Define $h:=\psi \circ g_m$: it is Morse with the same critical points and indices, since ${\mathrm{Hess}}_{p}h={\psi }^{\mathrm{\prime }}(g_m(p)){\mathrm{Hess}}_p{g}_m$ with ${\psi }^{\mathrm{\prime }}>0$, it equals $g_m$, hence $f$, near $\mathrm{\partial }W$, and $h(p)=d_{\mathrm{ind}p}=(\mathrm{ind}p+1)\mathrm{/}(n+2)$ at every critical point.

By the derivative-one choice in [F4], $h$ is $g_m$ plus a constant near every critical point. The old exact Morse coordinates and field model therefore remain valid for $h$, while $dh(X_m)={\psi }^{\mathrm{\prime }}(g_m)d{g}_m(X_m)<0$ at every regular point. Take $X^{\mathrm{\prime }}=X_m$, with its existing complete collar carrier and boundary directions.

The pair $(g,X^{\mathrm{\prime }})=(h,X_m)$ is adapted, has the original critical points and indices and boundary function, and satisfies $g(p)=(\mathrm{ind}p+1)\mathrm{/}(n+2)$. The simultaneous form of Interior slab handle attachment gives one handle per point, with all handles of each index at that common level. The empty critical set uses the identity reparametrization and the given pair. The final field is gradient-like for $g$ and also for the initial $f$, by the smallness invariant and the unchanged local models.