Convergence and composition of Morse tangle integrals

In the degree-$m$ ordered simplex, the set where some level $t_j=c$ has measure zero. Absolute convergence from [F2] allows removal of this set without changing the integral. The remaining domain is the disjoint union, for $k=0,\dots ,m$, of regions with exactly $k$ levels above $c$ and $m-k$ below it; each such region is the product of the corresponding ordered simplex for $T_2$ and that for $T_1$.

On the region indexed by $k$, every selected pair at a level above $c$ lies on $T_2$, and every pair below $c$ lies on $T_1$. Thus a pairing on the composite decomposes uniquely into a pairing on each piece. The product differential form separates into the upper and lower factors in height order, and the resulting chord diagram is their vertical stack.

Fubini's theorem applies to each product region because both factors are absolutely integrable by [F2]. Therefore its integral is the product of the degree-$k$ coefficient for $T_2$ and the degree-$(m-k)$ coefficient for $T_1$. Summing over the finite set $k=0,\dots ,m$ is exactly the degree-$m$ coefficient convolution of $Z(T_2)Z(T_1)$. Since $m$ was arbitrary, the displayed completed-series identity follows.