Let \(T_n\) be the lower-triangular prefix-sum matrix and let \(\cfrob(T_n)\) and \(\ctwo(T_n)\) be the factorization costs that govern mean and maximum per-coordinate squared error of the Laplace matrix mechanism under pure \(\eps\)-differential privacy, for \(\eps>0\). We prove \(\cfrob(T_n),\ctwo(T_n)=Θ\bigl((\log(n+1))^{3/2}\bigr)\) with no sign, sparsity, or squareness restriction and with arbitrary finite inner dimension. Consequently, within the pure-\(\eps\)-DP matrix-mechanism class the optimized maximum and mean squared errors are both \(Θ(\eps^{-2}\log^{3}(n+1))\). 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 Hölder's inequality transfers it to both factorization costs. The same computation determines \(\nucpow_p(T_n)\) 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-\(\eps\)-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.

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