The Geometry of Consistent Intervention in Layered 3D Worlds

Object-level edits to 3D worlds can break support and containment relations, while propagating an edit through the full dependency graph can move objects unnecessarily. We formulate translational intervention consistency in an object-position space where structural relations define a dependency-consistent feasible set. For a fixed support assignment, this set is a convex polyhedron induced by support equalities, support-surface inequalities, and containment constraints. We characterize how local interventions leave this feasible set and relate constraint violation to distance from feasibility. This geometry separates dependency from propagation: dependencies specify where an intervention may have consequences, while the intervention direction and local constraint geometry determine which relations actually require response. Consistency restoration is then formulated as a unique minimum-displacement projection within the appropriate feasible stratum, with stability guarantees under a fixed dependency structure. Experiments on reconstructed, RGB-D, synthetic, and generated 3D worlds evaluate prediction, propagation, repair, and consistency diagnosis. On InteriorGS, local geometry predicts intervention-induced inconsistency with AUROC 0.877, compared with 0.694 for graph fan-out. On a separate 340-edit benchmark, intervention-dependent propagation moves 59% fewer objects than full dependency closure while maintaining consistency in 334/340 edits. Independent physical rollouts, cross-dataset evaluations, and generated-world experiments further examine the relation between structural consistency and physical plausibility. Beyond post-hoc editing, the resulting feasible-state geometry provides an explicit consistency representation that can support verification and learning-time constraints in interactive and generative world models.