No 13-vertex triangulation of the three-torus

Kühnel and Lassmann constructed a neighborly triangulation of the three-torus on 15 vertices, and Lutz conjectured that it is vertex-minimal. The standard lower-bound arguments exclude at most twelve vertices but do not decide the 13- and 14-vertex cases. We exclude the 13-vertex case. Our proof uses the Novik--Swartz $g_2$-inequality and a simplicial-complement reduction to a nine-vertex model of the punctured three-torus. By analyzing fundamental groups of small simplicial complexes and applying Alexander duality in a vertex link, we obtain a contradiction from the deficiency of $\Z^3$.