Determination of the distance from a projection to nilpotents

In this note, we study the distance from an arbitrary nonzero projection $P$ to the set of nilpotents in a factor $\mathcal{M}$ equipped with a normal faithful tracial state $τ$. We prove that the distance equals $(2\cos \frac{τ(P)π}{1+2τ(P)})^{-1}$. This is new even in the case where $\mathcal{M}$ is the matrix algebra. The special case settles a conjecture posed by Z. Cramer.