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

This paper introduces the adversarial aggregation channel (AAC): a general framework in which a system must recover a correct collective decision from signals subject to adversarial corruption. The central objects are channel discriminability and adversarial capacity, and the central question is when consistent aggregation is possible despite strategic manipulation. The first main contribution is a Unified Impossibility Theorem proving that consistent aggregation fails if and only if the adversarial aliasing capacity meets or exceeds the channel's discriminability on at least one critical state pair. A Diagonal Intersection Principle reformulates aliasing geometrically: impossibility holds for a critical pair if and only if the adversary's reachable signal sets overlap in distribution space. Three sufficient conditions are developed — a support-function criterion reducing aliasing to a linear feasibility problem, a fiber dimension excess theorem, and a Chernoff reachability condition for product channels. Arrow's impossibility theorem, the Gibbard–Satterthwaite theorem, and the social-choice Nyquist boundary are shown to be instances of this single principle, with self-contained proofs of all three provided through the AAC framework. All three share a friction irrelevance property: the impossibility boundary is determined by informational capacity, not by costs imposed on the adversary. A Landauer principle for social choice formalizes this result: information destruction in the aggregation channel cannot be reversed by imposing costs on the process. The second main contribution is a quantitative conservation law for impossibility. When an impossibility at level k is resolved by enriching the evaluation channel with a verification mechanism, the discriminability gained decomposes exactly as: gain = meta-adversarial capacity + residual verification security. Under self-referential closure — meaning the verification mechanism is subject to the same class of adversarial corruption that produced the original impossibility — the meta-adversary fully corrupts the verification channel and the identity becomes exact. This conservation identity is verified across eight impossibility results spanning six domains: Gödel incompleteness, Turing undecidability, and Rice's theorem (mathematical 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). The quantum case extends the AAC from classical total variation to quantum trace distance. A No Free Lunch Theorem establishes that impossibility cannot be eliminated under self-referential closure: every enrichment that resolves an impossibility at one level creates at least an equal impossibility at the next. For product-channel enrichments, the conservation identity lifts to the exponential regime: the Chernoff information gained at level k exactly equals the KL budget required by the level-(k+1) adversary. The third main contribution is an Aggregation Channel Capacity Theorem — the AAC analogue of Shannon's noisy channel coding theorem. The aggregation capacity equals the number of critical pairs on which robust discriminability is positive. For product-channel AACs, a Net Capacity Formula decomposes the baseline Chernoff rate into adversarial rate consumed plus net aggregation rate, with the latter being the optimal error exponent. The Nyquist boundary is recovered as the condition at which the adversarial rate exhausts the baseline channel rate. The fourth contribution establishes an entropic foundation. A Second Law of Institutional Entropy proves that conditional Shannon entropy is non-decreasing across institutional levels under an institutional Markov property, with strict increase under self-referential closure. The institutional Markov property is characterized: it holds for sequential-processing architectures (where each level observes only the previous level's output) and fails generically for direct-access architectures (where each level observes the true state independently). Quantitative Landauer-type bounds connect entropy increase to the total-variation conservation quantities, and a degraded second law establishes robustness to bounded Markov violations. The fifth contribution derives a novel prediction: the Algorithmic Accountability Nyquist Boundary. 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 requirements, fairness tests, red-teaming — each fail under self-referential closure: the same model expressiveness that enables non-compliance enables gaming of any new audit metric. Constructive escape conditions are identified: monitoring channels that observe deployment-time behavior the designer cannot control (user complaint rates, hardware-level compute logs, cryptographically committed training data) break self-referential closure and permit genuine recovery of discriminability.