State-Changing Observation and the Decomposition of Retrieval Loss: An Instrument-Based Formulation of Loopback Observation Theory

This preprint extends the finite probabilistic model of Loopback Observation Theory (LOT) from nondisturbing record formation to state-changing observation, as a sequel to the FinStat-style model of probabilistic observation and retrieval hypotheses. A finite pre-observation state space, a finite record space, a finite post-observation state space, and a prior distribution are used to define a state-changing instrument, from which three prior-relative retrieval losses are constructed: from the record alone, from the post-observation state alone, and from both together. The paper does not claim to model the internal structure of the boundary M0, nor does it develop higher observer structures. The nondisturbing model of the preceding paper is recovered as a degenerate case in which the post-observation state coincides with the pre-observation state, and a post-state reset case, in which the post-observation state is independent of the pre-observation state, is identified as the opposite extreme. The joint retrieval loss is decomposed into the initial uncertainty about the pre-observation state, the information carried separately by the record and by the post-observation state, and a co-information interaction term. Under the sign convention adopted here, a positive interaction term indicates redundancy-like overlap, while a negative one indicates synergy-like behavior, in which the record and the post-observation state jointly retrieve more than the sum of their separate contributions. Output-separable instruments, for which the two outputs are conditionally independent given the pre-observation state, are shown to have a nonnegative interaction term and therefore cannot produce synergistic loopback closure. The mathematical components used here are standard: conditional entropy, mutual information, conditional mutual information, and co-information. The contribution of this paper is to organize these structures within the LOT triad of record formation, post-state realization, and retrieval, classifying state-changing instruments by whether pre-state information resides in the record, in the post-observation state, or in the coupling between them. AI-assisted: large language model tools (ChatGPT, Claude) were used for drafting, translation, and cross-checking; all claims and final content are the author's responsibility.