 Solution

Let $w$ be the primitive generator of the new ray $\tau$, and let $u,v$ be its adjacent primitive rays in the smooth subdivided fan. Smoothness gives a unique integer $a$ withBecause $\tau$ is new, it lies in the interior of an original two-dimensional cone. Its adjacent refined rays lie on opposite sides of it inside that cone, which forces $a>0$. Therefore $D_{\tau }^2<0$.