arXiv:2602.23613v2 Announce Type: replace Abstract: We propose an approximate interpolation construction and a practical coarsening algorithm for algebraic multigrid (AMG) method when applied to H(curl) problems. Building on the generalized AMG framework, we introduce an interior/exterior splitting that yields both a refinement-based and a fully algebraic construction of the interpolation. The refinement-based approach follows a geometric hierarchy, while the purely algebraic interpolation is constructed through a coarsening process that first coarsens a nodal dual problem and then builds coarse and fine variables using a matching algorithm. Under a uniform fine-space smoothing assumption, we prove a jump-independent two-grid convergence bound for canonical finite-element spaces. We also establish an exact commuting relation for the unshifted construction. Two-dimensional numerical experiments demonstrate sparse multilevel hierarchies and robust convergence under strong coefficient jumps in the curl--curl term. Three-dimensional experiments provide a two-level proof of concept, although the interpolation operators remain dense. Developing a practical sparse three-dimensional multilevel method is left to future work.