Uniform localization and boundary corrections for two-node quiver Coulomb branches

Abstract

We construct simple coherent classes and compute their normalized difference operators for the two-node quiver Coulomb branch with gauge group $GL_1\times GL_2$ and $m-1$ framing columns at the second node. The first magnetic charge is fixed at zero. For every $m\geq1$, every support $(n,-1)$ with $n\geq1$, and every integral electric twist, we prove that termwise saturation of the matter Koszul complex gives the simple intermediate extension. A graded normalization formula expresses its localization on every stratum as a finite binomial shuffle sum. The resulting operators satisfy bar symmetry, positive bubbling order, paired root-residue cancellation, and integral Laurent polynomial action. For $m=4$, we construct the derived boundary correction on $(2,-2)$, retaining the nonsemisimple odd-electric boundary object, and compute all electric twists. Finally, matter Euler divisibility proves triangular uniqueness on arbitrary fixed rank-two magnetic supports inside the actual integral Coulomb image. A pole-free counterexample shows why residue conditions alone do not give this uniqueness. Existence on general deeper mixed-sign supports remains open.