Multi-Channel Composition Theory for Adversarial Aggregation Channels: Separable and Correlated Adversaries

This paper develops the composition theory for adversarial aggregation channels (AACs) — the framework that unifies impossibility theorems across social choice, computability, mechanism design, and quantum information under a single conservation identity. While prior work treats individual AAC channels, this paper asks what happens when such channels compose, providing the AAC analogue of Shannon's classical composition calculus for cascade, parallel, and network topologies. For separable adversaries, four main results are established. The Cascade Composition Theorem decomposes total adversarial capacity across a series composition as Acasc = A₁ + A₂|₁ − L, where L ≥ 0 is a data-processing loss arising from the Dobrushin contraction coefficient, and the cascade aliasing set is shown to be monotone non-decreasing — cascading can only increase impossibility. The Parallel Composition Theorem establishes a maximum law for robust discriminability under independent observation. The Cascade Conservation Identity shows that the enrichment–corruption duality telescopes through cascade stages with an inter-stage adversarial synergy correction term. The Cascade Second Law proves that under the institutional Markov property, Shannon entropy of the true state is non-decreasing through cascade stages, with strict increase wherever self-referential closure occurs. The paper then extends the theory to correlated adversaries — multi-stage actors who can communicate and coordinate across institutional stages. A communication-constrained adversary model is introduced in which inter-stage correlation is mediated by a shared signal of bounded entropy R, interpolating between the separable and fully collusive regimes. The Correlated Adversary Cascade Theorem shows that correlated randomized adversaries can destroy discriminability that no single deterministic strategy can, via a cancellation mechanism impossible under independent randomization. The collusion premium Π (R) ≥ 0 measures the additional adversarial capacity gained from inter-stage communication and is bounded above by minR, Dʳobcasc. The collusion-distortion function — the adversarial analogue of Shannon's rate-distortion function — is introduced with a Lagrangian dual shown to be automatically convex, and strong duality is established away from finitely many phase-transition rates. The Binary Degeneracy Theorem proves that adversarial cancellation is a strictly higher-dimensional phenomenon: the collusion premium vanishes identically for binary signal spaces and is generically positive for |S| ≥ 3. For J-stage cascades, a multipartite collusion decomposition via inclusion-exclusion is derived with three-body corrections that are non-zero only when the adversarial communication graph contains cycles; when the topology is a tree, the pairwise decomposition is exact. This connects the AAC framework to the game-theoretic literature on communication equilibria (Forges, Barany, Myerson), with the collusion premium providing a quantitative bridge between mechanism design and information-theoretic impossibility. All results are verified against Shannon's classical composition theory as a consistency check, reducing to standard results in the non-adversarial limit.