Graph neural networks (GNNs) operate naturally on irregular reservoir grids, but low state error does not establish conservation, constitutive consistency, mesh transfer, physical interpretability, or numerical value. We audit a physics-certified framework for incompressible two-phase flow on unstructured finite-volume graphs. Pressure is exactly scalar-rate equivariant; a graph head predicts antisymmetric edge flux; and a discrete divergence projection enforces local continuity before conservative upwind transport. Projection reduces flux error by 47.1% - 57.1% and long-rollout saturation error by 73.6% - 86.1%, while driving non-well continuity defects from order unity to 10？14. A post-projection diagnostic nevertheless finds inverse-transmissibility-weighted Darcy defects of 0.27 - 0.59, proving that continuity does not imply constitutive compatibility. Under a strict matched budget (eight training reservoirs, 140 updates, ≈481.6 × 103 node-time presentations), multi-resolution exposure reduces projected-flux error by 42.0% - 73.2% at all five graph sizes; pressure gains are significant at three sizes and unresolved at two. In 24 multiwell reservoirs, hydraulic and tracer connectivity have mean injector-row rank correlation 0.632 at 0.5 PVI and top-producer agreement 73.6%. A tolerance-conditioned warm-start selector closely tracks the lower-iteration initializer, but reported savings are PCG iterations, not end-to-end runtime. The results support GNNs as structure-aware proposals coupled to discrete conservation and solver diagnostics rather than unconstrained simulator replacements.
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