Given any function $\tau\in C^{\infty}(\Delta)$ on a trivalent vertex $\Delta\subset D^{3}\subset\mathbb{R}^{3}$ with $\tau(0)=0$, there is a positive fibration whose base is locally affine isomorphic to the affine manifold with singularities $(\mathbb{R}^{3},\Delta_{\tau},\mathscr{A})$ of Example 3.11.