Coordinatewise Balanced Covering for Linear Gain Graphs, with an Application to Coset-List Min-2-Lin over Powers of Two

We study a list-constrained extension of modular equation deletion over powers of two, called Coset-List Min-2-Lin${}^{\pm }$ over $\mathbb{Z}\mathrm{/}{2}^d\mathbb{Z}$. Each variable is restricted to a dyadic coset $a+2^{\mathrm{\ell }}(\mathbb{Z}\mathrm{/}{2}^d\mathbb{Z})$, 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 $2^{-O(k^{2}r)}$, 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 $\rho$. 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 $$2^{O(k^2\rho +k\log (k\rho +2))}\cdot n^{O(1)}+\tilde{O}(md+{\rho }^{\omega }),$$ and the same framework returns a minimum-weight feasible deletion set among all solutions of size at most $k$.