Odd Euler characteristic for flag-no-square four-manifolds

We construct a closed connected PL four-manifold with odd Euler characteristic that admits a finite flag-no-square triangulation. Here flag means that every clique is a simplex, and no-square excludes induced four-cycles in the graph. The construction identifies antipodal points on the boundary of a deleted vertex star. In a regular hyperbolic tessellation, each graph edge outside that vertex changes the radial direction by at most $\pi/5$; this prevents the identification from creating a short nonclosing lift. A twisted double makes both the manifold condition and the global flag-no-square property explicit.