Solution
= 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 $a$ with
$$
u+v=aw,
\qquad D_\tau^2=-a.
$$
Because $\tau$ 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 $D_\tau^2<0$.