Consistency of Reconstruction Under Refinement: Filtered Spaces, Invariant Measures, and Exclusion Rules

A reconstruction from finite descriptions must coordinate refinement, observable equivalence, admissible transformations, exclusion rules, and statistical weights. We study when these operations remain consistent in one filtered system. The additional results concern exclusion and repair. An exact conditioning identity separates leakage into excluded coarse states from distortion among retained states. For punctured residue vector spaces at squarefree moduli, we classify all probability repairs invariant under invertible linear transformations under reduction to a proper divisor: the repair set is an explicit product of simplexes, while full support compatibility is impossible. The corresponding discrepancy has a finite orbit mass formula, and separately feasible divisor repairs can be jointly incompatible. These results use standard conditional probability, Chinese remainder decomposition, and orbit theory; the elementary identities are not claimed as new general principles. The established inverse limit, ultrametric, and invariant measure results provide the supporting framework and involve direct comparisons to symbolic dynamics, projective gauge constructions, and hierarchical probability. The worked models give quantitative limiting weights, and two published physical constructions illustrate the distinction between a geometric capacity response and a compatible probability read-out. The framework certifies conditional consistency and identifies incompatible interpretations without selecting application-specific assumptions.