How do separate minds come to share a world? Human cognition is recursive: we model the world; we also model how others model it, and how they model our modeling of it. This article presents Recursive Observer Entanglement (ROE), an integrative framework that treats shared reality as an emergent property of recursive predictive coupling among cognitive agents, and introduces two deliberately distinct constructs: Entanglement Depth (ED), a depth-weighted, time-integrated measure of how strongly two agents' reciprocal recursive predictions co-vary, and Observer Coherence (OC), an instantaneous, entropy-normalized measure of how aligned their current interpretive states are. Earlier statements of ROE specified ED and OC but never computed them, relying on a linear opinion-dynamics surrogate. Here that gap is addressed directly: in a minimal recursive-prediction model of two artificial agents, ED is computed as the damped sum of mutual information between reciprocal predictions and OC as an entropy-normalized ratio, both estimated from the agents' actual predictive distributions. Three results follow. First, ED and OC are well-defined, measurable, and both increase with coupling, with deeper recursion layers contributing progressively less. Second, a test of ROE's phase-transition prediction yields partial support: coherence depends on coupling structure in a way a linear model cannot reproduce, but the transition observed here is graded rather than sharp. Third, ROE's central distinguishing prediction is supported, in a design that removes an earlier confound: two agent ensembles with identical go-forward coupling, subjected to an identical shock and matched on post-shock coherence, differ only in the length of their pre-shock history and hence in accumulated ED (1.39 versus 0.02 bits). The high-ED ensemble recovers baseline coherence within three steps; the low-ED ensemble, coupled just as strongly, needs roughly an order of magnitude longer. ROE remains a framework rather than a validated theory, and human and neural validation are untouched, but the information-theoretic core is now operational in at least one working system.

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https://zenodo.org/records/21726937