We prove that the $L^2$ CVP distance from a random short ring element to the log-unit lattice of $\Q(ζ_{2^k})$ converges to $\fracπ{2\sqrt{6}}\sqrt{n}$ as $n=2^{k-1}\to\infty$. We then show that this target lies inside the Voronoi cell of the origin for $k\ge 4$. For the $L^\infty$ norm, the maximum over $n$ sub-Gaussian coordinates yields $O(\sqrt{\log n})$ which translates into a sub-polynomial approximation factor for the Short Generator Problem. We show a Coarse Lattice Theorem that Babai's algorithm returns zero for all structured targets, yet exactly recovers unit perturbations of arbitrary size. For module determinant ideals, we further prove the Trigamma Theorem that proves an intrinsic imbalance $σ_{g_0}=O(1)$ independent of the modulus $q$. Finally, combined with Parts I and II, we reduce the CDPR factor for ML-KEM from $\exp(\tO(\sqrt{n}))$ to a sub-polynomial value.

翻译：我们证明，从随机短环元到$\Q(ζ_{2^k})$的对数单位格的$L^2$范数CVP距离当$n=2^{k-1}\to\infty$时收敛于$\fracπ{2\sqrt{6}}\sqrt{n}$。随后证明，对于$k\ge 4$，该目标位于原点Voronoi胞腔内。对于$L^\infty$范数，$n$个次高斯坐标的最大值产生$O(\sqrt{\log n})$，这转化为短生成元问题的次多项式近似因子。我们证明了一个粗格定理：Babai算法对所有结构化目标均返回零，但能精确恢复任意大小的单位扰动。对于模行列式理想，我们进一步证明Trigamma定理，表明存在与模数$q$无关的内在非平衡性$σ_{g_0}=O(1)$。最后，结合第一、二部分，我们将ML-KEM的CDPR因子从$\exp(\tO(\sqrt{n}))$降低至次多项式值。