The Defendability Bounds: Architectural Conditions of Governability under Generative Adversaries

The series' deposited papers locate the frontier of detection: the content channel collapses operationally under budgeted mimicry [Rouxel 2026a, Hypothesis 3.12, Theorem 3.13], the temporal escape closes [Rouxel 2026b, Corollary 4.2, Corollary 4.4], durable resistance rests on three mission-relative anchors [Rouxel 2026c, Theorem 4.1, Propositions 5.1a–5.1c], deceptive guarantees decompose under disclosure [Rouxel 2026d, Proposition 3.1, Theorem 6.4], and the governability of a deployed system is bracketed by two bounds and an undecided band [Rouxel 2026e, Theorem 4.5]. Each result judges a deployed system at its coordinates; this paper makes their displacement its object. An architecture is an operator on a defended system preserving its substrate and its mission, and defendability, indexed by an anticipated threat model $\mathbb{T}$ and a declared load bound L, is the existence of an architecture whose deployed system is governable at the operating point: a design-time potential, where governability is a runtime state. The deployed constraint is a mediator, confining the state trajectory and signing the proposal trajectory, whose rejected attempts feed the loop with specification-fixed meaning. The main result lifts the governability bounds from deployed systems to substrate–mission pairs: the defendability regions $\mathfrak{D}_{\mathrm{in}}, \mathfrak{D}_{\mathrm{out}}$ and the band between them. The inner direction is an existential lifting with an exhibited witness, non-conjectural under its declared ledger: when the operator orbit meets the inner system class at an offered exclusion anchor covering $\mathbb{T}$, the pair is defendable by one witness uniform over the capability growth, which a certificate theorem certifies. The outer direction has two clauses at two statuses: an empty compatible anchor offer leaves the pair uncertifiable under every architecture, non-conjecturally, and, where no deployment rests on a floor the series sets aside, undefendable under the series' standing conjecture on non-anchored resistance with its premises. The band is declared undecided. Design corollaries follow: the violation channel; the incompleteness dilemma, every encoding of a rich mission leaking or amputating; the composition law, observational juxtaposition creating neither shield nor exclusion, separation requiring a joint-channel test, an unanchored joint buying temporization only. Under capability growth, the bounds separate what growth can displace, each coordinate under its named hypothesis, from the excluded states and the withheld information: an adversary preparing off target or acting through a legitimate component shifts its entry point and its attack surface, and stays barred from the states excluded by a constraint whose coverage remains certified at the endogenous threat index, which the architecture's own configuration may enlarge. Flow governance enters the lever inventory.