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

This paper introduces the adversarial aggregation channel (AAC)—a formal framework in which a decision-maker must recover a correct collective judgment from signals transmitted through a channel subject to adversarial corruption. The framework provides a unified, information-theoretic foundation for impossibility theorems across social choice theory, logic, computability, mechanism design, and quantum information. The central result is the Unified Impossibility Theorem: consistent aggregation fails if and only if the adversary's aliasing capacity meets or exceeds the channel's discriminability for at least one critical state pair. This principle is formalized geometrically via the Diagonal Intersection Principle (DIP)—impossibility holds precisely when the adversary's reachable signal sets for a critical pair overlap in distribution space. Arrow's impossibility theorem, the Gibbard–Satterthwaite theorem, and the social-choice Nyquist boundary are derived as instances of the DIP. The Gibbard–Satterthwaite theorem is derived directly from fiber dimension excess without invoking Maskin monotonicity or the Muller–Satterthwaite theorem. A Characterization Theorem identifies Decision-Channel-Corruption (DCC) structure as the necessary and sufficient condition for faithful AAC embedding, and proves that the Hausdorff–Banach–Tarski paradox and Vitali non-measurability theorem are not embeddable. The Enrichment–Corruption Duality proves an exact conservation identity: when an impossibility is resolved by enriching the channel with a verification mechanism, the discriminability gained decomposes as Δ = A_{k+1} + D^rob_V, where A_{k+1} is the meta-adversary's capacity to corrupt the enrichment and D^rob_V is its residual security. Under self-referential closure, conservation is tight (A_{k+1} = Δ) and a No Free Lunch Theorem shows total impossibility is non-decreasing across enrichment levels. The identity is verified across eight impossibilities spanning six domains: Gödel's incompleteness theorems, the halting problem, Rice's theorem, Arrow's theorem, the Gibbard–Satterthwaite theorem, the Myerson–Satterthwaite theorem, the social-choice Nyquist boundary, and quantum no-cloning. Additional results include: an Aggregation Channel Capacity Theorem (the AAC analogue of Shannon's noisy-channel coding theorem), a Second Law of Institutional Entropy (conditional entropy of the true state is non-decreasing across institutional levels, with strict increase under self-referential closure), a Friction Irrelevance Theorem characterizing when the aliasing boundary is invariant to cost perturbations, and a Landauer Principle for Social Choice showing that information destroyed by aggregation cannot be recovered by imposing costs. A novel Algorithmic Accountability Nyquist Boundary demonstrates that sufficiently expressive algorithm classes cannot be audited regardless of sample size or penalties. A General SRC Theorem eliminates the need for case-by-case self-referential closure verification. The paper closes with a dynamic adversarial channel theory in which robust discriminability satisfies a flow equation, is a supermartingale under stochastic self-referential closure, and undergoes an institutional collapse phase transition at the dynamic Nyquist boundary.