3D scene graph prediction is commonly optimized with object-node and relation-edge classification losses. These objectives encourage exact object and relation label recovery, but they do not directly optimize semantic proximity to the ground-truth scene graph: two wrong graphs can receive comparable penalties even when one is much closer in object roles and relation contexts. A direct graph-distance loss is difficult in general, since graph matching and graph edit distance require correspondence search and edit-cost design. Our key observation is that prior supervised 3D scene graph prediction protocols are conducted on fixed object-node indices and supervised edge slots. This can turn a scene graph comparison into a well-defined aligned graph-distance problem. To this end, we propose SWORD, a Scene-Grounded Wasserstein Object-Relation Distance that builds block-normalized object and relation ground distance metrics from training scene graph annotations through a deterministic base-to-context order, without external semantic priors. SWORD lifts these metrics to Wasserstein distances over aligned node and null-augmented edge marginals, providing semantic graph-distance supervision beyond exact-label penalties while preserving the ground-truth graph as the unique zero-distance solution in aligned marginal probability space. We prove that SWORD is a strict metric on aligned marginal probabilistic scene graphs, preserves scene-graph-grounded substitution ordering, controls additive graph-grounded semantic observables, and yields a surrogate upper-bounding decoded semantic error.

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