A linear gap for the maximal cp-rank

We prove that the maximal cp-rank pₙ satisfies pₙ ≥ n(n−3)/2 for odd n ≥ 5 and pₙ ≥ n(n−3)/2 − 1 for even n ≥ 6. Together with the known upper bound pₙ ≤ n(n+1)/2 − 4, this gives pₙ = n²/2 + O(n). Version 2 (28 September 2026) adds a citation to Hildebrand's order-five copositive family (Linear Algebra Appl. 437 (2012) 1538–1547). The results and proofs are unchanged.