The Conservation Identity Is Forced: An Axiomatic Characterization of the Adversarial Aggregation Channel, with a Quantum Extension

This paper proves that the Adversarial Aggregation Channel (AAC) conservation identity Δ = 𝒜V + DʳobV is the unique quantitative extension of Arrow's decisive-coalition framework. The result is established in two stages. Stage 1 (Classical Forcing): Five axioms — Operational Discriminability, Adversarial Corruption, Data Processing, Exact Bookkeeping, and Boolean Recovery — jointly determine the AAC. A Categorical Uniqueness Theorem shows that the two-term decomposition is the unique exact decomposition whose terms are natural transformations in the category of adversarial channels (AdChan) ; any finer decomposition requires a non-natural selection rule on the adversary's action space. Stage 2 (Quantum Forcing): The five axioms extend to quantum channels by replacing total variation with trace distance (forced by the Helstrom bound) and classical adversaries with CPTP maps. The quantum conservation identity is categorically forced by the same argument; the no-cloning theorem is derived as the quantum Self-Referential Closure (SRC) condition. The full forcing chain is: Arrow → qualitative AAC → quantitative AAC → quantum AAC → no-cloning as SRC. The five axioms are independent. The categorical uniqueness theorem and quantum forcing are new results.