The Governability Region: A Theory of the Defensive Inferential Loop under Generative Adversaries

The deposited papers of this series take the defender's position apart arc by arc: the content channel collapses operationally under budgeted generative mimicry [Rouxel 2026a, Theorem 3.13], the observation channel can only contract separation and can create the collapse [Rouxel 2026b, Theorem 3.2, Theorem 3.5], durable resistance organizes around three defensive anchors [Rouxel 2026c, Propositions 5.1a–5.1c], and defensive deception instruments them [Rouxel 2026d, Corollary 4.9]. This paper defines the object those results presuppose, the defensive inferential loop of observation, inference, decision, and actuation, and asks under what conditions the whole still closes. Two results carry it. The Closure Criterion (Theorem 3.3) is an assembly theorem at an operating point (α, β) over declared classes of adversary processes: a sufficient condition at a net margin, worst case over each class, and a necessary condition against the grid member, both on a good-decision event carrying an effect predicate, with an undecided band between them and, under the named independent-grid hypothesis, a pointwise biconditional at each fixed class against the independent grid member, with no uniform quantitative margin claimed; the response tempo τᵣ enters as a fourth temporal primitive, endogenous to the loop and adversarially targetable. The governability region bounds (Theorem 4.5) define mission-relative governability, loop closure conjoined with uniformly resistant coverage of the mission's requirements, and bracket its region: an inner bound, non-conjectural under a declared ledger of named hypotheses, where a closing policy with anchor-certified coverage exists; an outer bound with three separately routed obstructions, informational at one class of the growth, temporal under the uniform response floor and the grid clause, anchor-side conditional on [Rouxel 2026c, Conjecture 5.2] and on the requirement-level realized cofinality (H-RCOF)ᵣ at some requirement r; and between them a band the theory does not decide, with no threshold, monotonicity of the region, or critical surface established. Two extensions read the bounds under motion. Adversary→apparatus channels are treated as displacements of the region's coordinates, the ordering established on the direct channel and not on the closed-loop effect. A locally stationary pathwise reading lifts uniform governability to local governability along an interaction path, factors its first loss into a closure exit and per-requirement coverage exits, and routes by coordinate the window an adversarial policy may compress: invariant under a persisting exclusion, envelope-bounded on the attempt clock under a withheld secret, path-sensitive elsewhere; it is an interpretation for agentic policies and asserts the compression of no training procedure. An ancillary appendix gives the attainable knowledge supremum a fixed-point model under named axioms; no conjecture of the corpus is discharged.