Exact Rank of the Largest Good Lattice Point Matrix

We prove that for every integer $n>2$ the largest good lattice point matrix $U_n=[u_h:h\in H_n]$, whose columns are generated by all integers $h<n$ coprime to $n$, has rank exactly $φ(n)/2+1$. The upper bound follows from the pairing identity $u_h+u_{n-h}=(n,\dots,n,2n)^{\top}$. For the lower bound