The Impossibility Functor: A Categorical Unification of Impossibility Theorems

This paper constructs the category Imp of impossibility contexts—adversarial aggregation channels equipped with self-referential closure—and proves that a single functor, the impossibility functor F: Imp → Poset, classifies all impossibilities arising from adversarial signal corruption. The functor assigns to each object an impossibility gap κ = 𝒜 − D^rob, the signed deficit between adversarial capacity and robust discriminability; impossibility holds if and only if κ ≥ 0 at some critical pair. Four main theorems are established. The Functorial Conservation Theorem identifies the conservation identity Δ = 𝒜_V + D^rob_V as the unique natural transformation between the capacity functor and the self-referential demand functor on Imp. The Propagation Theorem shows that impossible objects form a coideal under morphisms. The Meta-Impossibility Theorem proves that the framework predicts its own incompleteness under self-referential closure. The Full Structural Separation Theorem establishes that the social choice tower (Arrow, Gibbard–Satterthwaite, Myerson–Satterthwaite) and the logic tower (Gödel, Turing, Rice) are categorically disconnected, formalizing the folklore intuition that "Arrow's theorem and Gödel's theorem are different kinds of impossibility" as a precise disconnection theorem. This separation is resolved by showing that the Nyquist object is a universal source reaching every other object in Imp, while the logic tower is a terminal sink. The construction subsumes the eight impossibilities previously unified by the adversarial aggregation channel framework and extends it in two directions. A ninth object—the Fischer–Lynch–Paterson (FLP) impossibility of asynchronous consensus—is constructed and categorically separated from all social choice objects. A new falsifiable result, the Verification Insufficiency Theorem, is derived: no binary (pass/fail) audit mechanism can eliminate strategic manipulation in any social choice function with ≥ 3 alternatives, regardless of accuracy or computational power, because a binary audit has rank 2 while the Gibbard–Satterthwaite adversary has rank ≥ 3. Additional structural results include the Gap Defect Formula, the adversarial rank functor, the Minimum Compensation Theorem, and a 2-categorical formulation in which the No Free Lunch Theorem appears as a fixed-point property under self-referential closure.