When a neural time-series model reports that one variable modulates another's effect on a target, is the discovered interaction a property of the data or an artifact of model flexibility? We argue that this is fundamentally a question of identifiability, governed by the geometry of the observed input support rather than by the specific neural architecture. We study the problem in a multiplicative-gating extension of neural additive vector autoregression (GNAVAR), in which source contributions are modulated by other lagged variables. We show that representational capacity is not identifiability: dependent inputs induce leakage between edge-specific interaction terms, and low-dimensional support permits distinct interaction decompositions that agree on the observed data while differing elsewhere. We then prove a population identifiability theorem for normalized minimal GNAVAR decompositions under explicit support conditions, including settings with shared modulators. The theory yields a simple practitioner-facing diagnostic: the effective rank of the joint lag-block covariance predicts, before fitting, whether interaction recovery is feasible for a given candidate set. When the candidate set is unknown, a two-seed stability check provides a practical operational test. The same support condition organizes empirical outcomes into the three states predicted by the theory. Our results show that interaction recoverability depends on support geometry, that effective rank provides a practical pre-fit diagnostic, and that instability across independent fits is a characteristic signature of non-identifiable interaction discovery. The identifiability phenomenon, the support condition, and the instability signature are model-agnostic; GNAVAR is the vehicle that makes them provable.