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 $\varepsilon$-differential privacy, for $\varepsilon>0$. We prove $c_{\mathrm{F}}(T_n),c_2(T_n)=\Theta((\log(n+1))^{3/2})$ with no sign, sparsity, or squareness restriction and with arbitrary finite inner dimension. Consequently, within the pure-$\varepsilon$-DP matrix-mechanism class, the optimized maximum and mean squared errors are both $\Theta(\varepsilon^{-2}\log^3(n+1))$. 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 $D_k(T_n)\asymp n^{3/2}k^{-1/2}$, valid in the low-rank range $1\leq k\leq n/16$, 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 $\mathfrak{n}_p(T_n)$ for each fixed $0<p<1$: order $n$ below $2/3$, $n\log n$ at $2/3$, and $n^{3p/2}$ above. A Fenwick interval factorization supplies matching upper bounds. The claims are confined to pure-$\varepsilon$-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.