A proof that the maximum cp-rank in order six is nine

The paper proves that every real completely positive matrix of order six has
an entrywise nonnegative factor with at most nine columns, and exhibits a
positive definite completely positive matrix of order six with cp-rank nine,
so that the maximum cp-rank in order six is nine. The argument uses the
published order-five bound and Shaked-Monderer's corrected theorems for
matrices orthogonal to exceptional extremal copositive matrices. A
relative-face shear reduces a putative counterexample to three support graphs:
$K_{3,3}$, the triangular prism and the octahedron. The central additional
lemma gives an eight-column bound for octahedrally supported matrices
orthogonal to a copositive matrix with positive diagonal. Proper octahedral
subgraphs are then bounded by eight, and one rank-one subtraction bounds the
full octahedral graph by nine, including singular endpoints. Exact finite
checks supplement the analytic proof.