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. 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. 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). 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. 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. Quantitative Landauer-type bounds connect entropy increase to the total-variation conservation quantities, and a degraded second law establishes robustness to bounded Markov violations.