Trading concentrates a simply connected h-cobordism in two adjacent middle indices

By [F1] take an ordered presentation with indices at least two and at most $n-1$. Each incoming inclusion is a homotopy equivalence and the relative homology vanishes by [F2].

Eliminate the indices $q=2,\dots ,k-1$, not the desired surviving index $k$. At each stage $N_q={\mathrm{\partial }}_1{W}_q$ is simply connected: the later handles have index at least $q+1\ge 3$, so ${\pi }_1(W_q)\cong {\pi }_1(W)=1$, and the reversed trace from $N_q$ has only index $n+1-q\ge 4$ handles, so ${\pi }_1(N_q)\cong {\pi }_1(W_q)$. These are the handle fundamental-group comparisons of the belt-complement helper in [F4]. With no indices below $q$, one has $C_{q-1}=0$ and ${\mathrm{\partial }}_{q+1}$ is onto $C_q$. For a fixed $q$-handle $e$, choose coefficients $y_j$ with ${\sum }_j{y}_j{\mathrm{\partial }}_{q+1}[{\phi }_j]=[e]$. Start with a trivial framed $q$-sphere of class zero and apply [F3] to obtain $\beta$ of class $[e]$, isotopic to that trivial sphere one level higher. Use the framed version of [F3], starting with the standard trivial frame, so $\beta$ is framed-isotopic to that trivial sphere in the next level.

Since $q\le k-1\le n-3$, [F4] applies to $\beta$. Its $q=2$ incoming condition follows from simple connectivity of $M_0$, or equivalently from the h-cobordism fundamental-group isomorphism. Carry the higher attaching embeddings along the resulting ambient isotopy; its induced diffeomorphism of the next level preserves standard framed triviality. A diffeomorphism maps a standard framed disk embedding to another such embedding. By [F5] trade $e$ for one $(q+2)$-handle. Finite repetition eliminates all original indices below $k$.

Reverse the triad and perform the same low-index eliminations through dual index $n-k-1$. Every such index is at most $n-3$; the opposite incoming face is simply connected, so the same $q=2$ condition holds. Trading a dual $r$-handle for a dual $(r+2)$-handle introduces original index $n-r-1\ge k$, and thus never recreates an original handle below $k$. This removes all original indices at least $k+2$ while preserving the prior low elimination. The surviving original indices are exactly $k,k+1$, for the full stated $2\le k\le n-2$ range.

The surviving relative chain complex is concentrated in degrees $k,k+1$ and has zero homology by [F2]. Its differential is injective and surjective, giving equal finite handle numbers and the stated two-term isomorphism. The normal form retains every advertised index without invoking the unproved arbitrary-sphere flipped endpoint.