Suppose $C$ transduces all graphs. Then it produces every powerset bipartite graph, so the edge formula of the transduction shatters arbitrarily large vertex sets in colored members of $C$: a set $W$ and, for every Boolean pattern on $W$, a realizer whose trace on $W$ is that pattern. Uniform quasi-wideness of a nowhere dense class finds inside $W$, after deleting a set $S$ of at most $s$ vertices, a subset of any requested size that is pairwise $3^(q+2)$-independent in $G − S$. Color every vertex by the rank-bounded local type of its decorated ball in $G − S$, the decorations recording the colors and the adjacency to $S$; there are boundedly many types, so a pigeonhole yields $s + 4$ scattered vertices of one type. The realizers of $s + 2$ chosen traces are pairwise distinct, so one of them avoids $S$, and the ball-swap lemma forces it to be $3^q$-close to one of the first two scattered vertices and to one of two later ones — two scattered vertices at distance at most $2·3^q < 3^(q+2)$, a contradiction.

The quasi-wideness input is not reproved here: it is assumed from the Sparsity Lectures submission, whose nowhere-denseness definition is the very one this statement is phrased over, so no transport is needed and $Lax5Proofs.QuasiWideness$ only reshapes the conclusion from $Set$ to $Finset$. Everything below it — shattering, local types, ball swapping and the pigeonhole — is proved here.

The theorem is Adler and Adler's, Interpreting nowhere dense graph classes as a classical notion of model theory (European Journal of Combinatorics, 2014), who prove the stronger monadic stability. The proof here is the deletion specialization of the flip-breakability route: uniform quasi-wideness and a semantic locality argument, with rank-bounded local types of decorated balls and a ball-swap back-and-forth in place of Gaifman's theorem.