Gauge-Invariant Local Geometric Relations for Constructible Scene Semantics

A geometric scene may admit many coordinate descriptions, primitive parameterizations, and construction histories. These freedoms obstruct direct alignment of geometry with engineering meaning: the transverse frame of an unmarked cylinder is invisible to its solid, while an attached port or child may depend on that frame. We develop a hierarchical method that anchors a local chart to a candidate parent, compensates every dependent quantity under chart changes, extracts frame-independent typed relations, and matches complete local patterns across scenes. A business interpretation is valid precisely when the intended business state is constant on the fibers of the extracted relation; otherwise the method must retain candidates or abstain. We connect this criterion to an evidencepreserving PHG-style construction tree and a CCS-based geometric layer, while separating canonical representation of a selected state from recognition from reduced observations. Lean 4 verifies the gauge-equivalence and hierarchical compensation laws, the relationto-business factorization criterion, a restricted constructive normalization example, and an exact axis-aligned planar rod-pair profile. In 139 native-mesh Blender scenes from a plant demonstration, 3,347 route meshes yield 854 axial box candidates and 524 resolved near-pair motifs. Twelve coarse motif keys recur across scenes; 437 planar pairs are imported into Lean and all pass the declared extraction checks. These are geometric motif results, not certified pipe connections or business-role accuracy. The contribution is a precise criterion and executable research path for reference-independent local semantics, with explicit conditions under which objective geometric relations can support engineering interpretation.