Costs of Arbitrary Real Matrix Factorizations for Pure-DP Continual Counting

Let T n be the lower-triangular prefix-sum matrix and let c F(T n) and c 2(T n) be the factorization costs that govern the mean and maximum per-coordinate squared error of the Laplace matrix mechanism under pure epsilon-differential privacy, for epsilon>0. We prove c F(T n),c 2(T n)= (((n+1)) 3/2) with no sign, sparsity, or squareness restriction and with arbitrary finite inner dimension. Consequently, within the pure- epsilon-DP matrix-mechanism class, the optimized maximum and mean squared errors are both (epsilon-2 3(n+1)). Under the factorization contract of Arkhipov and Kalinin (arXiv:2607.08963v1), who prove the matching lower order for factors with entries in 0,1 and state the arbitrary-factor extension as open, the theorem below establishes the order for arbitrary real factors. The lower bound runs through a p-nuclear obstruction: an aggregate column-width estimate D k(T n) n 3/2k-1/2, valid in the low-rank range 1 at most k at most n/16, for the prefix chain, fed into the classical approximation-space conversion of Pietsch and Hinrichs--Pietsch, becomes harmonic at the critical exponent p=2/3, and H"older's inequality transfers it to both factorization costs. The same computation determines n p(T n) for each fixed 0<p<1: order n below 2/3, n n at 2/3, and n 3p/2 above. A Fenwick interval factorization supplies matching upper bounds. The claims are confined to pure- epsilon-DP Laplace matrix mechanisms and the two stated squared-error criteria; they do not cover non-matrix continual mechanisms, approximate-DP sensitivity, or expected maxima across coordinates.