Three-Stage Lexicographic Allocation on Directed Irrigation Trees with Conveyance Losses and Capacity Constraints: A Closed-Form Stage-1 Fairness Guarantee and Operator–Balance Equivalence

Water allocation in lossy canal trees must respect local capacities and a service guarantee. We develop a deterministic three-stage lexicographic model for a rooted, fixed-route tree: Stage 1 maximizes the minimum service ratio, Stage 2 maximizes weighted seasonal satisfaction subject to that guarantee, and Stage 3 minimizes temporal service-ratio variation while preserving both optima. The nonnegative packing structure yields the Stage-1 optimum in closed form from capacity-to-full-load ratios, and the loss-aware path operator equals node balance. Across 400 randomized instances, 140 with capacities not tied to their own loads, and 60 more with load-independent scales, the closed form agreed with linear programming (LP) to 2.2 × 10−16, confirming every bottleneck the formula named. On 500-user trees, Stage 1 took 12.1 ms against 93.1 ms for the LP. Stage 3 lowered the temporal variation only where multiple Stage-2 optima were detected. Of five allocation rules, only the hierarchy, equal proportional allocation, and full leximin always preserved the guarantee; of those three, the hierarchy delivered the most water in the median. In a controlled Gone Abat Jap scenario, ±10% demand perturbations moved the binding resource from source to reach in 61% of draws. The results verify a model-specific characterization, not field performance.