Can We Know the Whole from Within? The Embedded Observer Problem and the Limits of Global-State Identification

Abstract

We study global-state identification by an observer embedded within a system and restricted to finite-budget, geometrically connected observations. The central question is whether information that is collectively sufficient to identify a global state can necessarily be integrated by a single admissible observer. For finite-field linear models, we characterize accessible information through rooted connected supports and the cycle space of the underlying graph. This leads to an exact characterization of the collective identification threshold in terms of rooted cycle access. For an n × n square grid rooted at a corner, we prove that the collective identification threshold is 2n + 2, whereas complete integration of the cycle space by a single rooted observer requires 2n(n + 1) observations. Consequently, the ratio between the single-observer and collective thresholds is exactly n and therefore grows without bound with system size. The result establishes an explicit separation between collective global identifiability and single-observer integration. We additionally derive an asymptotic integration-gap bound and extend the framework to additively noisy observations on finite Abelian groups, where the geometric separation admits a corresponding interpretation through Blackwell comparison of statistical experiments.