From Primitive Observation Records to the Dual-Axis Axiom: Set-Theoretic Foundations of Observational Geometry

Abstract

The problem considered here arose in information science: the relation that forms an observation need not remain fixed while the underlying state of interest is held fixed. Yet the argument begins before any particular information-theoretic, statistical, geometric, or physical model is imposed. Its primitive object is a totalized observation table µ0 : P×A → Y⊥, where ⊥ records nonimplementability. Two kinds of variation are kept separate from the outset: a change in recorded value on the implementable domain and a change in implementability itself. If two formation operations give different table entries at the same object label, no single-axis map that preserves that object-label identity can represent the complete table. Quotienting the object and formation labels by complete row and column behavior then produces two effective state sets, an admissible joint domain B ⊆ X × S, and an observation law F : B → Y. The object-side quotient is only a behavioral state relative to the observation family. When an independent physical theory supplies a physical state space whose identity must be preserved, an explicit compatibility interface retains that space and minimizes only the formation side. For the resulting family of observation maps, every complete behavior-minimal realization is uniquely isomorphic to the canonical one. If two implementable formation states in the same first-state fiber yield different outputs, no unary recoding that preserves the first-state projection can identify them. The second effective state is thus fixed, up to behavior-preserving isomorphism, by the role of selecting the complete observation-formation map; only then is it called the observer state. The resulting two-role structure is summarized by the dual-axis axiom of observation as a retrospective interface to the preceding results, not as a premise in their derivation. Conditional, label-free, and joint identifiability are then distinguished, and the fiber product B ×X B is identified as the canonical set-theoretic domain for comparison at fixed first state. No probability law, topology, differentiable structure, metric, or group action is required. The construction isolates a pre-statistical and pre-geometric layer of observation formation while keeping behavioral reduction, physical identity, and identifiability distinct.