On discrete Twist and Four-Flux in N=1 heterotic/F-theory compactifications

We give an indirect argument for the matching $G^2=-{\pi }_{\ast }{\gamma }^2$ of four-flux and discrete twist in the duality between N=1 heterotic string and $F$-theory. This treats in detail the Euler number computation for the physically relevant case of a Calabi-Yau fourfold with singularities.