After having defined a 3-strings midpoint-inserted vertex for the bc system, we
analyze the relation between gh=0 states (wedge states) and gh=3 midpoint duals. We find
explicit and regular relations connecting the two objects. In the case of wedge states this
allows us to write down a spectral decomposition for the gh=0 Neumann matrices, despite
the fact that they are not commuting with the matrix representation of K1. We thus trace
back the origin of this noncommutativity to be a consequence of the imaginary poles of
the wedge eigenvalues in the complex -plane. With explicit reconstruction formulas at
hand for both gh=0 and gh=3, we can finally show how the midpoint vertex avoids this
intrinsic noncommutativity at gh=0, making everything as simple as the zero momentum
matter sector.
