The Authorization Imperative: Why statistical alignment cannot close the Authorization Gap, and why a non-writable, fail-closed gate can

Autonomous AI systems are moving from generating text to executing actions in physical, digital, and financial domains. Training, test-time search, and moderation can improve proposal quality, but none of them, by itself, establishes whether a proposed action is authorized to execute. This paper calls that missing function the Authorization Gap. We make two contributions. First, an illustrative quadratic model in which, at any finite boundary stiffness, every positive conflict produces boundary movement, while making the boundary immutable still leaves a positive residual violation. Second, a general account of execution authorization that does not depend on how the proposer works: universal authorization holds exactly when every reachable protected execution is authorized (Theorem 1); any reachable unauthorized path defeats it (Corollary 3); an independent, sound, action-bound, default-deny mechanism is sufficient (Theorem 2); and unresolved authorization uncertainty at a final decision requires withholding permission (Theorem 3). We present Predictive Loop Control, a four-layer architecture whose authorization layer is specified to meet Theorem 2, with a worked funds-transfer example. Review draft, circulated for technical criticism.