Sharp-Order Closed-CPSD Separation for Odd-Prism Context Matrices

An explicit family of doubly nonnegative matrices is constructed from four-element contexts on odd prism graphs. For every odd integer n ≥ 5, the matrix K_n has size 6n × 6n, rank 16, unit diagonal, and pairwise distinct Gram vectors. It satisfies the prescribed context orthogonality and normalization identities. The main theorem determines the sharp asymptotic order of the distance from K_n to the unit-diagonal slice of the closure of the completely positive semidefinite (CPSD) cone. In the maximum-entry norm, this distance is bounded above and below by positive constants times 1/n², with constants independent of n. The result holds for both real and complex PSD factorizations. The lower bound is uniform over all factor dimensions and ranks. Its proof combines a common Naimark dilation, weighted Hilbert–Schmidt estimates, and the odd-cycle spectral gap. The matching upper bound is realized by explicit real 8 × 8 positive semidefinite factors of rank two. Together, these estimates establish the optimal decay exponent. The accompanying verification package contains the manuscript source, explicit construction data, and scripts for exact symbolic verification and numerical replay.