Finite-Horizon Constrained Selection Operators for Hidden-State Observation

We study hidden-state observation for finite-dimensional constrained systems in which any admissible readout must be selected from the finite-horizon reachable set of the current state. The construction is formulated as a constrained-observer problem: a candidate observed state must be reachable under the declared update family and must satisfy all internally generated channel constraints. The central object is the admissibility-intersection core IC (X) = R_Δ (X) ∩ ⋂₈∈₈ Cᵢ, where R_Δ (X) ⊆ D_Ω is the finite-horizon reachable set and the Cᵢ are channel-feasibility sets. A selected state is obtained by the minimum-burden rule XC = argminZ∈IC (X) BC (Z; X), followed by an internally generated rank-m readout Π^ (m) ₒbs (X; C) = Q^ (m) C (XC). The case m = 3 is recovered as the three-coordinate readout relevant to downstream physical observation channels. We prove existence of the selector under compact-regularity and finite-intersection compatibility, uniqueness under strict admissible convexity, and resolved observation under a local rank-m branch condition. The construction enforces a non-generation property: X*C ∈ R_Δ (X), so the observer cannot assign a state outside the system's declared finite-horizon reachable set. It also induces three mutually exclusive regimes: infeasible observation (empty intersection), resolved observation (unique selector with full-rank readout), and unresolved observation (nonunique selection or rank failure). The framework does not require probabilistic, Hilbert-space, or external measurement-postulate assumptions. It is intended as a mathematical observation-interface layer for constrained systems, closely related to reachable-set methods, set-membership estimation, and moving-horizon estimation, but distinguished by the simultaneous use of reachable-set non-generation, internally generated channel constraints, minimum-burden selection, and rank-certified readout. We further extend the framework to bipartite admissible channel pairs and define the joint correlation functional via Cesàro averages. A channel-context function is introduced: it maps each admissible channel to its selected admissible representative at a fixed hidden state. Distinct selected representatives under distinct local channels establish channel-sensitive selection at the observer layer. This fact alone does not establish failure of shared-antecedent factorization and does not remove the Bell–CHSH bound. We prove only that an independently supplied shared-antecedent factorization with binary outputs implies |S| ≤ 2; nonfactorizability or a Bell-violating channel requires a separate certificate and remains open.