Balanced cut-vector distance formula ($\Vert v_X-v_Y{\Vert }_2^2=4t(2p-t),t=|X\cap Y|$)

Represent a balanced partition of $[4p]$ by the zero-one vector recording which unordered pairs lie across its cut. Complementary subsets represent the same cut; choose the representative containing coordinate one. Two such cuts differ on the pairs crossing the symmetric difference of their sides, giving the displayed distance. Its maximum is $4p^2$, attained exactly when $t=p$. Every cut has $4p^2$ occupied coordinates, so the vectors lie in an affine hyperplane of dimension at most $\left(\genfrac{}{}{0ex}{}{4p}{2}\right)-1$.