We study a list-constrained extension of modular equation deletion over powers of two, called Coset-List Min-2-Lin$^{\pm}$ over $\mathbb{Z}/2^d\mathbb{Z}$. Each variable is restricted to a dyadic coset $a+2^{\ell}(\mathbb{Z}/2^d\mathbb{Z})$, each binary constraint is of the form $x_u=x_v$, $x_u=-x_v$, or $x_u=2x_v$, and the goal is to delete a minimum number of constraints so that the remaining system is satisfiable. This problem lies between the no-list case and the poorly understood fully conservative list setting. Our main technical result is a coordinatewise balanced covering theorem for linear gain graphs labeled by vectors in $\mathbb{F}_2^r$. Given any balanced subgraph of cost at most $k$, a randomized procedure outputs a vertex set $S$ and an edge set $F$ such that $(G-F)[S]$ is balanced and, with probability $2^{-O(k^2r)}$, every hidden balanced subgraph of cost at most $k$ is contained in $S$ while all incident deletions are captured by $F$. The proof tensors the one-coordinate balanced-covering theorem of Dabrowski, Jonsson, Ordyniak, Osipov, and Wahlström across coordinates, and is combined with a rank-compression theorem replacing the ambient lifted dimension by the intrinsic cycle-label rank $ρ$. We also develop a cycle-space formulation, a cut-space/potential characterization of balancedness, a minimal-dimension statement for equivalent labelings, and an explicit bit-lifting analysis for dyadic coset systems. These yield a randomized one-sided-error algorithm running in \[ 2^{O(k^2ρ+k\log(kρ+2))}\cdot n^{O(1)}+\widetilde{O}(md+ρ^ω), \] and the same framework returns a minimum-weight feasible deletion set among all solutions of size at most $k$.

翻译：我们研究模方程在二幂次上的带列表约束的删除扩展问题，称为 $\mathbb{Z}/2^d\mathbb{Z}$ 上的陪集列表最小二线性模式（Coset-List Min-2-Lin$^{\pm}$）。每个变量限制为二进陪集 $a+2^{\ell}(\mathbb{Z}/2^d\mathbb{Z})$，每个二元约束形如 $x_u=x_v$、$x_u=-x_v$ 或 $x_u=2x_v$，目标是最小删除约束数量，使剩余系统可满足。该问题介于无列表情形与尚未充分理解的完全保守列表情形之间。我们的主要技术成果是面向 $\mathbb{F}_2^r$ 向量标记的线性增益图的坐标化平衡覆盖定理。给定任意代价不超过 $k$ 的平衡子图，一个随机化过程输出顶点集 $S$ 和边集 $F$，使得 $(G-F)[S]$ 平衡，并且以概率 $2^{-O(k^2r)}$，每个代价不超过 $k$ 的隐藏平衡子图均包含于 $S$，而所有关联删除由 $F$ 捕获。证明通过坐标张量扩展了 Dabrowski、Jonsson、Ordyniak、Osipov 和 Wahlström 的单坐标平衡覆盖定理，并结合秩压缩定理，将环境提升维数替换为内蕴的循环标记秩 $\rho$。我们还发展了循环空间表述、平衡性的割空间/势刻画、等价标记的最小维数陈述，以及二进陪集系统的显式比特提升分析。这些成果给出一个随机单侧错误算法，运行时间为 \[ 2^{O(k^2\rho+k\log(k\rho+2))}\cdot n^{O(1)}+\widetilde{O}(md+\rho^ω), \] 且同一框架可返回所有大小不超过 $k$ 的解中代价最小的可行删除集。