Path identities in a rigid quiver representation

A base-change invariant polynomial identity holding on a rigid representation holds on its dense orbit and hence throughout the representation space. A path through nonzero vertex spaces cannot vanish identically there: assign every involved arrow a rank-one map carrying a chosen source vector to the chosen target vector. This works for repeated arrows. Consequently a nonzero path image in a rigid representation cannot be killed by an outgoing arrow into a nonzero vertex space.