A Categorical Derivation of the Dual-Axis Structure of Observation

A complete observation may change because the prepared-object condition changes or because the physical process forming its record changes. We ask when the latter dependence becomes an irreducible state responsibility. The primitive input is an operationally role-faithful closed-world table μ0 : P × A → Y_⊥. Here the object and record-forming roles are certified by experimental protocol or antecedent physics, rather than inferred from the recorded values, and ⊥ denotes certified nonimplementability. If two operations give different entries at one fixed object label, no exact encoding that preserves the certified first-role map can identify the witnessed pair. Quotienting complete row and column behaviour removes precisely duplicate labels and yields reduced behavioural objects X and S, a descended law eF : X × S → Y⊥, and an admissible joint domain B. The second quotient is canonically bijective with the implemented family H_F ⊆ Map(X , Y⊥), thereby acquiring a behaviour-minimal observer-state identity. Record-forming non-triviality makes S non-singleton, while injectivity of the exponential transpose forbids any further factorwise lossless quotient. The reduced realization is terminal among active, product-separable, pointwise-faithful realizations and is invariant under admissible role-preserving empirical covers, including redundant refinements and relabellings. These results do not assert that every arbitrary joint encoding admits a global S-decoder; such a decoder is an additional compatibility condition. For a fixed reduced interface, whether the observer state is fixed, recorded, hidden or actively controlled changes later inference, not the complete conditional law. We state the resulting interface retrospectively as the Dual-Axis Axiom of Observation. Within the operationally role-faithful domain, it expresses a foundational structural law: witnessed same-first-role dependence cannot be merged or made to factor through the first role alone without changing the complete observational content or abandoning the certified first-role semantics.