Vertex Labelling of a Half-Cube to Induce Fibonacci Face Labels

Label the ten vertices of a hexagonal half-cube with digits 0, 1, 2, …, 9 (using each digit exactly once) such that the induced labels of the six faces (other than the hexagon), given by the sums of the labels of the vertices around those faces, are all Fibonacci numbers. We want a complete list of all solutions to the puzzle using mathematical logic, without a complete search of all permutations of the digits. The purpose of this paper is to inspire logical reasoning.