The Natural Dual-Axis Structure of Observation: Categorical Foundations and Consequences for Quantum Theory
Abstract
Physics ordinarily writes an observation as y = h(x), locating change in the state x while leaving the conditions that form the fact in the background. We prove that, whenever those conditions vary at fixed certified first-state identity, the complete law possesses a natural and irreducible dual-axis structure. The primitive datum is an operationally role-faithful table µ0 : P × A → Y⊥, with ⊥ denoting certified nonimplementability and with no probabilistic, linear, or quantum structure assumed. A fixed-first-role witness blocks every exact factorization through the first projection. Coequalizing identical complete rows and columns then produces behavioral state objects X and S, a unique law Fe : X ×S → Y⊥, and its feasible joint domain. This realization is terminal among active typed exact realizations: every competitor maps uniquely onto it, and the fibers of that map are exactly the competitor’s behavioral redundancies. Moreover, S is canonically bijective with the realized family of complete observation maps, admits no further lossless factorwise quotient, and is invariant under role-preserving empirical covers. When antecedent physics supplies a compatible physical-state map, the construction lifts canonically to Fephys : Xphys × S → Y⊥, yielding the Physobser Axiom. Classical laws are its fixed-observation slices. Quantum theory realizes the two axes as preparation state and observation event, with the Born pairing as their interface; contextuality and Bell…
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