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

We introduce the adversarial aggregation channel (AAC): a decision-theoretic system that must recover a correct collective decision from signals transmitted through a channel subject to adversarial corruption. Each AAC has a channel discriminability and an adversarial aliasing capacity. The paper establishes five main contributions. Unified Impossibility Theorem. Consistent aggregation is impossible if and only if the adversarial aliasing capacity meets or exceeds the channel's discriminability for at least one critical state pair. Arrow's impossibility theorem, the Gibbard–Satterthwaite theorem, and the social-choice Nyquist boundary are all instances of this single principle. A Diagonal Intersection Principle (DIP) reformulates impossibility as a geometric condition—intersection of reachable signal sets in distribution space—with three sufficient conditions (support-function criterion, fiber dimension excess, Chernoff reachability). A Characterization Theorem identifies the exact scope of the framework: an impossibility admits a faithful AAC embedding if and only if it possesses Decision-Channel-Corruption structure, with the Hausdorff–Banach–Tarski paradox and Vitali non-measurability exhibited as concrete non-examples. Conservation Law (Enrichment–Corruption Duality). When an impossibility is resolved by enriching the evaluation channel with a verification mechanism, the discriminability gained decomposes exactly as Δ = A + D_rob, where A is the meta-adversary's capacity to corrupt the verification mechanism and D_rob is its residual security. Under self-referential closure, A = Δ: the impossibility migrates completely. This formalizes a conservation conjecture articulated informally by Patty (2026). The identity is verified across eight impossibility results spanning six domains: Gödel incompleteness, Turing undecidability, and Rice's theorem (logic and computability); Arrow's theorem, the Gibbard–Satterthwaite theorem, and the Myerson–Satterthwaite theorem (social choice and mechanism design); the Nyquist boundary (institutional inference); and the quantum no-cloning theorem (physics). A No Free Lunch Theorem guarantees impossibility persists across all enrichment levels. Aggregation Channel Capacity Theorem. The AAC analogue of Shannon's noisy channel coding theorem. The aggregation capacity—the maximum number of critical pairs simultaneously resolvable with vanishing error—equals the number of pairs with positive robust discriminability. For product-channel AACs, a Net Capacity Formula gives the optimal error exponent on each critical pair as the baseline Chernoff rate minus the adversarial rate consumed. Second Law of Institutional Entropy. The conditional Shannon entropy of the true state is non-decreasing across institutional levels under an institutional Markov property, with strict increase under self-referential closure. Quantitative Landauer-type bounds connect entropy increase to total-variation conservation quantities, with robustness to bounded Markov violations. Algorithmic Accountability Nyquist Boundary. A novel prediction: 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, compliance costs, or penalties. The conservation law proves that sequential audit enrichments (explainability, fairness tests, red-teaming) each fail under self-referential closure. Three constructive escape conditions are identified.