Non-Eliminability of Observation Conditions: Dynamical Closure and Cross-Scale Retention

Abstract

Abstract

The same object state can produce different responses under different observation-formation conditions. We ask which information must be retained once the identities of the two sides and the response task have been specified independently, and how this requirement changes with the prediction horizon and the scale of description. Constancy of the response on fibers characterizes joint sufficiency and the eliminability of either side. Including identity in the output allows the standard behavioral quotient construction to yield the coarsest closed state representation that preserves the prescribed identities. Nested tasks under a common evolution induce a compatible family of predictive scales. Exact closure, however, does not ensure that observation conditions remain non-eliminable after coarse-graining: the roles of both sides and a response difference must also survive. A model of two coupled chains gives explicit expressions for the required initial state and the fraction of a response difference retained by a mode projection; a nonuniform chain illustrates the failure of closure. Independent calibration and held-out evaluation use synthetic data, and their numerical results agree with the full classical theory of coupled dynamics. The contribution is a unified analysis of identity constraints, predictive closure, and the retention of response differences, supported by reproducible model calculations. No new theory of minimal realization, or new physical effect implied by the dual-axis structure alone, is claimed.