Is there something on the reachability problem but with the assumption that at all steps you must have \\( \leq d_i \\) of object \\( X_i \\)? This is breaking the symmetric monoidal structure by taking a very small subset of the objects, but I wanted a finite dimensional linear operator on \\( \otimes_i \mathbb{C}^{d_i+1} \\) that sends each computational basis state to the result of applying any one of the reactions (including identity) in uniform superposition (provided still satisfy all the constraints). Then apply this some large number of times and evaluate the matrix element between starting and ending to see if it is 0. I see some with \\( d_i =1 \\), but not anything higher.