Finite-Width Renormalisation of Dihedrally Programmed Multi-Weyl Nodes

Finite transverse width does not alter the symmetry-programmed release order of the dihedral multi-Weyl hierarchy, but it does renormalise the microscopic response tensor supporting the topological node. For scalar curved quantum waveguides with D2q symmetry, we derive an O(ell^2) Kato-Feshbach description that separates direct finite-width corrections from external-state dressing and reduced-resolvent contributions. Across q = 2,...,8, the geometric response becomes increasingly sensitive to virtual dressing as the programmed charge increases, while the Aharonov-Bohm flux mass remains overwhelmingly direct-dominated. Independent full finite-width tubular calculations at q = 3, 4, and 8 reproduce the reduced response. A charge-eight convergence audit further shows that the Chern number and branch conditioning can appear converged before the underlying octic tensor coefficient itself has converged. The results distinguish rigid topological charge from width-sensitive local quantum geometry and establish a finite-width response framework for dihedrally programmed multi-Weyl nodes.