Unconditional Subexponential Arithmetic Circuits\ for Positive Factorial Multiples

Let $τ(n)$ be the least number of binary additions, subtractions and multiplications required to construct $n$ from $1$ and $2$, with reuse of intermediate values. We show that, for every sufficiently large integer $k$, some positive integer $m$ satisfies \[ τ(mk!)\leq \exp\!\left(O\!\left((\log