Finite Directional Coefficient Modules over Euler Fibers of $A$-Hypergeometric Systems

Let \(A\) be a full-rank integer matrix, let \(β\) be a complex parameter, and consider the associated \(A\)-hypergeometric system. Without assuming homogeneity, pointedness, or positivity, we study directed formal logarithmic solutions over all exponent lattices in the Euler fiber. For a fixed lattice-generic direction, negative-support strata are described by rational sign polyhedra, and normalization reduces the coefficient equations on each exponent lattice to a finite directional coefficient module. We prove that only finitely many exponent lattices contribute nonzero directed solution spaces and that all contributing coefficient modules have finite length. Macaulay inverse-system duality identifies the dimension of the all-lattice solution space with the length of the corresponding all-lattice module. We also give an explicit formal-series realization and a polynomial colon-ideal criterion for all realizable lowest exponents.