Lemma: A finite bound for the table probe

Every contact entry in (eq:encoding) has absolute value at most $768$.

Proof. Only species words with two occurrences of each species contribute. For each such word there are four compatible external assignments. The two differentiated momenta each have two nonzero components. The tensors have respectively $3,2,2,2$ nonzero entries, so at most

component products occur per assignment. Each product has absolute value at most $2$. Matching constraints can only remove terms. Thus $4\cdot 96\cdot 2=768$ is a valid bound.