Artificial intelligence is diffusing through organizations faster than many firms can redesign the structures that direct, monitor, coordinate, and learn from its use. This formal theory-building article develops \emph{AI governance deficit} as an organizational-level, directional misfit between governance demand generated by AI deployment and effective governance capacity. The construct is positioned explicitly against dynamic capabilities, absorptive capacity, institutional pressure, responsible-AI maturity, and prior uses of ``governance deficit'' at the global-regime level. Rather than treating deficit as a difference score or as a desirable absence of rules, we locate it on a response surface that preserves the independent levels and joint configuration of demand and capacity. We distinguish passive discretion created by governance lag from authorized autonomy deliberately produced by enabling governance. A semi-formal model derives the conditions under which a conditional inverted-U can arise, when it collapses into a monotonic negative relation, and why persistent misfit is more damaging than transitory misfit of the same instantaneous magnitude. Unresolved governance obligations accumulate as control debt with cognitive--epistemic, structural--coordination, and institutional--relational manifestations. The framework integrates governance architecture into the core function and specifies falsifiable response-surface, longitudinal, and ecosystem-level tests.