Positivity of simplicial volume and uniqueness for the twelve-vertex flag three-spheres

We study the twelve-vertex flag triangulation $T_{12}$ of $S^3$ and the Davis $4$-manifold $M(T_{12})$ associated with it. We prove that $T_{12}$ is, up to isomorphism, the unique flag triangulation of $S^3$ with twelve vertices in which every edge is contained in a square; this proves Milizia's Conjecture~6.31. We also prove a general criterion for the tangent Euler class of a Davis $4$-manifold to be bounded, and we verify it for $T_{12}$ by means of an explicit symmetric form. Since the Euler class takes the value $256$ on the fundamental class of $M(T_{12})$, the duality between bounded cohomology and simplicial volume gives $\|M(T_{12})\|>0$. This answers Milizia's Question~1.7.