Observation Map Adjunction and Retrieval Ambiguity: A Minimal Model for Loopback Observation Theory

This preprint presents Version 1.1 of “Observation Map Adjunction and Retrieval Ambiguity: A Minimal Model for Loopback Observation Theory.” The paper develops a deliberately minimal finite-set model for Loopback Observation Theory (LOT). Here, “minimal” does not mean a universal or unique minimality claim. It means that the model is restricted to finite sets, a single deterministic observation map q: Ω → R, powerset posets, and the direct-image/inverse-image adjunction needed to express record formation and record-side re-expansion. Given a finite latent state space Ω, a finite record space R, and an observation map q: Ω → R, the direct image F(S)=q[S] and the inverse image U(T)=q⁻¹[T] satisfy F(S)⊆T if and only if S⊆U(T), so that F⊣U. In this finite-set model, the boundary structure M0 of LOT is represented by this adjunction. The composite UF induces the observational closure cl_q(S)=UF(S)=q⁻¹[q[S]], which closes a state-side set along the fibers of q. For nonempty S⊆Ω, the retrieval-ambiguity increase is defined by loss_q(S)=log₂(|UF(S)|/|S|). This quantity is not a Shannon-information quantity, but a finite-set cardinality-based measure of how much the candidate state set expands after record formation and record-side re-expansion. On the counit side, the composite FU satisfies FU(T)=T∩Im(q), which identifies the attainable part of a record-side set T. The paper does not claim to introduce new order-theoretic structures. The mathematical components are standard from the viewpoints of Galois connections, closure operators, and rough set theory. The contribution is to reposition these known structures as a deliberately minimal finite-set representation of M0, organizing state-to-record coarsening, record-side re-expansion, retrieval ambiguity, and record attainability within LOT.