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We propose a fully quantitative theory for the finite-size scaling of the filling transition in a three-dimensional double wedge geometry, based on the exact transfer-matrix solution of a phenomenological interfacial model. Antisymmetric fields act at the top and bottom wedges; so each one favours a different bulk phase under coexistence conditions, i.e. gas and liquid phases in fluid models such as the lattice gas, or equivalently ferromagnetic domains of opposed magnetization in the Ising model. From this formalism we obtain an analytical form for the magnetization probability distribution function at critical filling which is valid for any aspect ratio. To test our predictions we revisit and perform new simulation studies of filling in the Ising model double-wedge geometry and use our finite-size scaling theory to locate accurately the critical filling transition.