  We identify a reasonably large class of pushouts of strict $n$-categories
which are preserved by the "inclusion" functor from strict $n$-categories to
weak $(\infty,n)$-categories. These include the pushouts used to assemble from
its generating cells any object of Joyal's category $\Theta$, any of Street's
orientals, any lax Gray cube, and more generally any "regular directed CW
complex." More precisely, the theorem applies to any \emph{torsion-free
complex} in the sense of Forest -- a corrected version of Street's \emph{parity
complexes}.
  This result may be regarded as partial progress toward Henry's conjecture
that the pushouts assembling any non-unital computad are similarly preserved by
the "inclusion" into weak $(\infty,n)$-categories. In future work we shall
apply this result to give new models of $(\infty,n)$-categories as presheaves
on torsion-free complexes, and to construct the Gray tensor product of weak
$(\infty,n)$-categories.
  This result is deduced from an \emph{$(\infty,n)$-categorical pasting
theorem}, in the spirit of Power's 2-categorical and $n$-categorical pasting
theorems, and the $(\infty,2)$-categorical pasting theorems of Columbus and of
Hackney, Ozornova, Riehl, and Rovelli. This says that, when assembling a
"pasting diagram" from its generating cells, the space of "composite cells
which can be pasted together from all of the generators" is contractible.
