Completion Probabilities in Finite Public-Service Referral Models with Uncertain Routing

A public-service referral route can be available while its probability of completion remains unspecified. This paper develops a finite, fixed-policy probability enrichment that preserves that distinction. Normalized transition rows define a path law over a time-bounded referral graph. A route-refinement example shows the sensitivity of uniform route counts to descriptive choices. Standard hitting-probability recursion and a finite derivation of the Doob h-transform explain how conditioning on completion changes observed route shares. For uncertain routing, independently selectable compact row sets support backward lower and upper recursions; a shared-parameter example shows how rectangular relaxation can widen the bounds through incompatible local choices. A fictional referral dossier connects these results to missing terminal records, admission denominators and participant-defined outcomes. Verified probability and robust dynamic-programming sources supply mathematical antecedents. The proposed contribution concerns the assumptions needed for interpretable completion estimates. Empirical calibration, causal policy effects, interacting service capacity and the normative value of completion remain open.