The Ontological Dominance Papers Theorem 4: The Adversarial Extension

This paper states Theorem 4 of the Ontological Dominance Papers, the Adversarial Extension, governing deliberate, coordinated efforts by competitors to damage an entity's AI representation. The theorem formalizes competitive noise floor decomposition under adversarial conditions, identifies temporal depth and vocabulary sovereignty as the two structural defenses that adversarial signal construction cannot overcome, and specifies a three-condition detection pattern distinguishing adversarial displacement from ordinary competitive erosion. Consistent with the Adversarial Displacement Theorem presented in OD-S, the central operational finding is that defensive structures must be constructed before an adversarial campaign begins, since temporal depth cannot be manufactured retroactively during an active attack. Theorem 4 was classified VERDICT B upon adversarial review: a formally correct extension of OD-1, confirmed theoretically, with empirical field confirmation pending execution of the designated falsification test.