Closed-Relational Reconstruction Systems: Viability, Observable-History Lifting, and Finite Obstruction Certificates

A many-to-one observation can create a fictitious history by joining transitions that require incompatible hidden states. We study this problem for compact Hausdorff stage spaces with closed continuation relations and closed observational equivalences. Compactness identifies indefinite viability with viability at every finite horizon and makes every nonliftable infinite observable history detectable by a finite prefix. Exact successor congruence (ESC), a stage-dependent bisimulation condition, guarantees lifting from every initial representative. We also give a strictly weaker necessary-and-sufficient criterion for existence of some lift, formulated using reachable sets of compatible representatives. In finite time-homogeneous systems, a shortest obstruction has at most ∑y(2|q−1(y)|−1) transitions, making exhaustive verification finite. Arbitrary finite Markov kernels descend exactly when their aggregate probabilities agree within each source class; uniform successor counting is a special case. A complete partially observed control example computes viability, detects a spliced plan, repairs the abstraction, and verifies a nonuniform kernel. A separate common-action proposition distinguishes existential realizability from admissibility for every retained candidate. The contribution is a unified history-level audit and its explicit criteria and certificates, built on classical inverse-limit, bisimulation, lumpability, and subset-construction methods. Physical reconstruction and Reconstruction Before Control motivate the framework; their empirical validity remains outside these model-relative guarantees.