Adversarial Aggregation Channels: Unified Impossibility, Conservation, and the Enrichment–Corruption Duality

This paper introduces the adversarial aggregation channel (AAC): a unified information-theoretic framework in which a decision-maker must recover a correct collective judgment from signals subject to adversarial corruption. The central objects are channel discriminability and adversarial aliasing capacity; impossibility holds if and only if the adversary can render two states informationally indistinguishable. The framework yields four main contributions. First, a Unified Impossibility Theorem and Diagonal Intersection Principle show that Arrow's impossibility theorem, the Gibbard–Satterthwaite theorem, and the social-choice Nyquist boundary are all instances of a single geometric condition—intersection of reachable signal sets in distribution space—with self-contained proofs of all three. Second, an Enrichment–Corruption Duality formalizes and proves Patty's conservation conjecture as an exact identity: discriminability gained by any channel enrichment decomposes as meta-adversarial capacity plus residual verification security. Under self-referential closure this identity is strict, verified across eight impossibility results spanning mathematical logic (Gödel, Turing, Rice), social choice (Arrow, Gibbard–Satterthwaite, Myerson–Satterthwaite), institutional inference (Nyquist), and quantum physics (no-cloning). A No Free Lunch Theorem follows. Third, an Aggregation Channel Capacity Theorem—the AAC analogue of Shannon's noisy channel coding theorem—characterizes the maximum achievable fidelity via a Net Capacity Formula: baseline Chernoff rate equals adversarial rate consumed plus net aggregation rate. Fourth, a Second Law of Institutional Entropy and an information-theoretic Landauer principle convert the thermodynamic analogy into theorems via the data processing inequality. As a novel prediction, an Algorithmic Accountability Nyquist Boundary is derived: when an algorithm class's obfuscation capacity meets or exceeds the audit channel's Chernoff information, no audit protocol can reliably detect non-compliance regardless of sample size, penalties, or compliance costs. A conservation law for sequential audit enrichment proves that explainability requirements, fairness tests, and red-teaming each fail under self-referential closure, with three constructive escape routes identified. The paper also develops a dynamic adversarial channel theory in which the adversary adapts in real time. A Dynamic Conservation Identity governs the evolution of robust discriminability; under stochastic self-referential closure the discriminability trajectory is a non-negative supermartingale converging almost surely. A Collapse Phase Transition identifies a critical adversarial dominance ratio separating institutional survival from collapse, and an Ergodic Conservation Theorem recovers the static framework as the equilibrium of the dynamic system.