Branching Networks as a Lawhood Problem: A Tier-0 Admissibility Resolution Beyond Steiner Optimization

Classical models of branching networks treat network geometry as a singular one-dimensional graph governed by length or volume minimization, with interior junction rules imposed as primitive stationarity conditions. While effective in idealized thin-link regimes, such models fail systematically in real physical networks, which exhibit higher-degree junctions, broad angle asymmetry, nonplanarity, and orthogonal sprouting across biological and engineered systems. This paper shows that the failure is structural rather than empirical. Using the Tier-0 admissibility framework for law selection, it is demonstrated that pure graph-based branching laws are not lawful fixed points once boundary canonicity, persistence under collapse, and closure under admissible completion are enforced. Singular Steiner-type object classes are excluded before optimization is considered. The analysis establishes that any admissible branching-network law must admit a smooth, persistence-stable, surface-dominated geometric completion. Minimal-area stationarity emerges as the unique local invariant extremal principle on this completion, not as an imposed modeling assumption but as a consequence of admissibility. Observed branching motifs are thereby reinterpreted as projections of the completed geometry rather than as singular graph rules. Independent empirical studies of physical and biological networks reporting surface-optimized local organization align with this structural conclusion. The result provides a law-level resolution of the branching-network discrepancy and demonstrates the explanatory power of closure-based admissibility criteria in a nontrivial geometric domain.