Pressure–Vorticity–Reaction Graph Networks: Structured Topological Memory and Safe Learned Dynamics

Learning constrained graph dynamics requires conservation, long-lived state, nonlinear response, and hard constraints. We introduce the Pressure–Vorticity–Reaction Graph Network (PVR-GN), which separates these roles. Pressure is a particular flow satisfying the current divergence; Vorticity is a persistent cycle-space state that stores topological memory without changing nodal conservation; and Reaction is a small causal proposal passed through a feasibility-preserving operator. We state the shared-budget allocation identity used by the cell, derive an allocation-free feasible set, and give fixed-active-set KKT sensitivities for the continuous projection branch. On a precommitted blind synthetic benchmark, PVR has zero observed rollout failures and favorable central comparisons with raw GNN/GRU baselines. A clustered reanalysis selects the 3,025-parameter PVR-Compact: its Blind-ID NMSE is lower than PVR-Strong by 0.001799 (95% CI [0.000526, 0.003124]); the pooled Blind-OOD difference is 0.000202 and its CI includes zero. In a separate control pilot, removing Vorticity or Reaction increases overloaded NMSE by 0.70706 or 0.02398; removing safe return yields 87.5% rollout failure under overload. A semantics-preserving same-A100 refactor reduces median paired Compact latency by 18.939%, although not uniformly. An independent static PGLib/MATPOWER DC-OPF study finds favorable held-out accuracy–compactness, numerically close scale transfer, and mixed rankings under severe congestion. In a separately frozen case57 test, a DC3-style DC-OPF adaptation (post-hoc) is natively feasible on both strata, while PVR-DC has lower dispatch MSE after common repair. PVR is a compact, auditable inductive bias for constrained graph dynamics, not a universal optimizer.