A short proof of the generalized Conway--Gordon--Sachs theorem

The famous Conway–Gordon–Sachs theorem for the complete graph on six vertices was extended to the general complete graph on $n$ vertices by Kazakov–Korablev as a congruence modulo $2$, and its integral lift was given by Morishita–Nikkuni. However, the proof is complicated and long. In this paper, we provide a shorter proof of the generalized Conway–Gordon–Sachs theorem over integers.