The non-self-crossing criterion for rigid modules over arbitrary acyclic quivers

Let $R$ be the path algebra of an acyclic quiver, let $g$ be the onto map from admissible curves to indecomposable modules corresponding to positive real roots, and let $M$ be an indecomposable $R$-module. The non-self-crossing criterion.

$$\operatorname{Ext}^1(M,M)=0 \quad\Longleftrightarrow\quad g^{-1}(M)\text{ contains a non-self-crossing curve}.$$

Thus rigidity of an indecomposable module should be detected by at least one non-self-crossing curve in its fiber under $g$. This is the proposed generalization from 2-complete quivers to arbitrary acyclic quivers; the supplied text gives no resolution.