Categorified Cluster Structures on Coulomb Branches of Certain Star-Shaped Quivers

We construct quantum cluster structures on a family of quantized $K$-theoretic Coulomb branches and prove that the associated loop-graded Koszul-perverse hearts give monoidal categorifications of these algebras. For each $r\ge 1$, the gauge group is the image of $G{L}_2\times ({\mathbb{C}}^{\ast }{)}^r$ in $GL(({\mathbb{C}}^2{)}^r)$, where $(g,z_1,\dots ,z_r)$ acts on the $a$th summand by $g{z}_a$. Each initial seed has $r+2$ mutable and $r$ invertible frozen variables. Over $\mathbb{Z}[v^{\pm 1}]$, the quantum cluster algebra coincides with its upper quantum cluster algebra and is isomorphic to the Grothendieck ring of the corresponding heart. Every quantum cluster monomial is represented by a simple object, and every mutation is realized by a short exact sequence. The proof uses a quadratic refinement compatible with mutation and explicit mutation sequences that produce the dressed monopole generators. The principal quiver has the four-punctured-sphere mutation type for $r=4$ and is mutation-infinite for $r\ge 5$.