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

Year: 2026
Type: preprint
Abstract: This paper develops a deliberately minimal finite-set model for Loopback Observation Theory (LOT). LOT is treated here as a working framework for describing observation in terms of a correspondence between a state-side description domain and a record-side description domain. In this model, a finite latent state space Ω, a finite record space R, and an observation map q: Ω → R are given. The powersets P(Ω) and P(R) are regarded as poset categories under inclusion. 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 is represented by this adjunction. The composite UF induces the observational closure cl_q(S)=UF(S)=q⁻¹[q[S]], which closes a set of latent states 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 cardinality-based finite-set measure of how much the candidate state set expands after record formation and record-side re-expansion. The paper proves the non-negativity of loss_q(S) and characterizes the zero condition as the case in which S is closed under the fibers of q. On the counit side, the composite FU satisfies FU(T)=T∩Im(q), which identifies the attainable part of a record-side set T. These constructions are not new order-theoretic results; they are well known from the viewpoints of Galois connections, closure operators, and rough set theory. In particular, cl_q(S) coincides with the upper approximation induced by the equivalence relation ω∼_qω′ if and only if q(ω)=q(ω′). The contribution of this paper is not to introduce a new order-theoretic structure, but to reposition these known structures as a deliberately minimal finite-set model of M0, state-to-record coarsening, record-side re-expansion, retrieval ambiguity, and record attainability within LOT.
DOI: 10.5281/zenodo.20702202
Language: English
Open Access status: green
OpenAlex ID: W7164843312
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