arXiv:2609.01469v2 Announce Type: replace-cross Abstract: Let $q$ be an odd prime. Let $\zeta_q$ be a primitive $q$th root of unity, and put $K=\mathbb{Q}(\zeta_q)$, with ring of integers $\mathcal{O}_K=\mathbb{Z}[\zeta_q]$. As $q$ ranges over odd primes, we prove that the decision version of the Shortest Vector Problem ($\mathsf{SVP}$) in the canonical $\ell_2$-norm is $\mathsf{NP}$-complete on full-rank free submodules of $\mathcal{O}_K^2$ by giving a deterministic polynomial-time many-one reduction from Exact Cover by 3-Sets (X3C). The module rank is fixed at two. The main obstacle is closure under the action of $\mathcal{O}_K$. A module containing a nonzero vector also contains every scalar multiple of that vector by a nonzero element of $\mathcal{O}_K$, and some of these multiples may be shorter. Three ideas overcome this obstacle. First, we map the Bennett--Peikert Reed--Solomon lattice to a principal cyclotomic ideal and use Wan's point-count estimates to prove that a coset of this ideal contains many binary coefficient representatives. Second, we construct a checker whose canonical squared norm encodes the X3C equations, with a controlled nonnegative error term. Third, the checker and a second module coordinate combine with a separation bound for ideal cosets to rule out short vectors in NO instances. Each constructed instance consists of an odd prime $q$, two integral generators whose $2\times2$ generator matrix has nonzero determinant, and an integer squared threshold. The construction also gives $\mathsf{NP}$-hardness of search-$\mathsf{SVP}$ under deterministic polynomial-time Turing reductions with at most one oracle call.