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

This work studies when an observer embedded in a finite graph can identify global constraint information while being restricted to connected, finite-budget observations. A rooted access theorem characterizes exactly which cycle constraints can be recovered collectively and yields a general characterization of the collective-identification threshold for arbitrary finite connected graphs, together with a corresponding single-observer threshold through the rooted cyclic hull. The main exact application is an a by b rectangular square-cell grid rooted at a corner. The collective-identification threshold is exactly a + b + 2, while the single-observer threshold is exactly 2ab + a + b. The square n by n case follows as a corollary and gives an exact separation ratio of n. More generally, the separation can grow without bound along rectangular families in which both side lengths increase. This demonstrates that information collectively available across an admissible observer class may require a substantially larger observation budget to be captured by any single admissible observer. The framework is also extended to common-noise additive experiments on finite Abelian groups. A subgroup and Fourier characterization of the Blackwell order is combined with an adaptive finite-budget criterion for the existence of a Blackwell-greatest admissible observer. The manuscript and supplementary material provide complete proofs, boundary-case analysis, dependency and scope checks, and exhaustive regression tests on small rectangular grids.