Orthogonal partition by four hyperplanes

We study orthogonal $(4,3)$-equipartitions of finite measures absolutely continuous with respect to Lebesgue measure: every three of four mutually orthogonal affine hyperplanes must divide each measure into eight equal parts. For every number $m$ of measures, we determine the exact dimension threshold of Musin’s polynomial criterion and express it in terms of the binary expansion of $m$. The threshold coincides with the geometric lower bound precisely when $m=2^j-1$. Consequently, ${\mathrm{\Delta }}^{\ast }(2^j-1,4,3)=7\cdot 2^{j-1}-2$ for every $j\ge 1$.