From these conjectures follows that the actual obstacle vanishes. Indeed, is a derivation of . (But not of the product!) Therefore, when is physical (i.e. ) we automatically have . But the kernel of on the ghost number three cohomology is trivial. Therefore the obstacle actually vanishes.
This explains how it could be that the equation for the dilaton is consistent for almost all deformations, but becomes inconsistent on nonphysical. (“How come the obstacles to solving for form a finite-dimensional space?”) We now see that nonphysical deformations form a separate branch.