The nonnegative rank of the nine-point linear distance matrix

We prove that the matrix $D_9=((i-j)^2)_{i,j=1}^9$ has nonnegative rank seven. The lower bound combines the polygon-section interpretation of restricted nonnegative rank with Sylvester's rank inequality applied to both factors. In a factorization with at most six terms, each factor would have ordinary rank at least five; their product would then have rank at least four, a contradiction. The known seven-term reflection factorization, written with integer entries, supplies the matching upper bound. More generally, every rank-three square matrix of order at least nine with zero diagonal and strictly positive off-diagonal entries has nonnegative rank at least seven.