Abstract
Abstract This paper establishes the quantitative vector-valued multi-indexed inequalities for multilinear fractional integral operators under the full range of the multilinear Muckenhoupt class. By constructing an explicit discretization of the multi-indexed potential kernel, the continuous operator is structurally circumscribed by localized dyadic averages over sparse families. This localized estimate facilitates a deterministic decoupling scheme that aligns with contemporary limited-range, off-diagonal extrapolation mechanisms. Consequently, we bypass the topological obstructions inherent in direct sequence-space dualization regimes and obtain quantitative vector-valued norm inequalities with an explicitly tracked dependence on the weight characteristic. The resulting estimates systematically isolate the geometric properties of the underlying operators near the integrability boundaries, providing structural transparency for potential-type scaling dynamics without relying on qualitative abstract limiting procedures.