The Maximal Finite CP-Plus-Rank Equals the Maximal CP-Rank

The cp-plus-rank of a symmetric matrix is the smallest number of columns in an entrywise strictly positive Gram factor. We prove that, in every order n, its largest finite value equals the maximal cp-rank p_n. The key observation is that a strictly positive factor remains strictly positive after a sufficiently small contraction along the Perron eigendirection. Combining this negative deformation with the positive Perron lift of Bomze, Dickinson and Still realizes every finite cp-plus-rank as an ordinary cp-rank in the same order and with the same ordinary matrix rank. The argument includes singular matrices and extends the corresponding interior two-sided Perron-ray identity. It determines the relation between the two maxima, not a numerical formula for p_n or the largest pointwise gap. The accompanying source archive contains a self-contained proof, audit notes and exact rational regression code. This preprint is self-audited and unrefereed; no absolute priority, independent review or formal-verification claim is made. AI-assisted tools supported research, computation, proof development and manuscript preparation. The author is responsible for the final text. Corpus identifier: OWR-15957-003.