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

Let $T_n$ be the lower-triangular prefix-sum matrix and let $c_{\mathrm{F}}(T_n)$ 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 $c_{\mathrm{F}}(T_n),c_2(T_n)=\mathrm{\Theta }((\log (n+1){)}^{3\mathrm{/}2})$ 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 $\mathrm{\Theta }({\epsilon }^{-2}{\log }^3(n+1))$. 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 $D_k(T_n)\asymp n^{3\mathrm{/}2}{k}^{-1\mathrm{/}2}$, valid in the low-rank range $1\le k\le n\mathrm{/}16$, for the prefix chain, fed into the classical approximation-space conversion of Pietsch and Hinrichs–Pietsch, becomes harmonic at the critical exponent $p=2\mathrm{/}3$, and H"older’s inequality transfers it to both factorization costs. The same computation determines ${\mathfrak{n}}_p(T_n)$ for each fixed $0<p<1$: order $n$ below $2\mathrm{/}3$, $n\log n$ at $2\mathrm{/}3$, and $n^{3p\mathrm{/}2}$ 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.