Cluster-Guided Topological Semantic Spectra: A Numerically Testable Framework for Local Geometric Semantics in Complex Engineering CAD Scenes

<title>Abstract</title> <p>Recovering engineering semantics from complex three-dimensional CAD scenes is often treated as primitive classification, object detection, or relation-graph recognition. These formulations are insufficient for industrial geometry because local shape and semantic identity are not in one-to-one correspondence: a cylinder may be a pipe, a handrail member, a support rod, a flange slice, or part of equipment; a ring-like elbow motif may be a valve handwheel, a guardrail corner, a lifting eye, or a decorative frame. This paper proposes a cluster-guided topological semantic spectrum for local geometric semantics. The central claim is not that geometry directly determines semantic labels, but that semantic recovery becomes numerically testable when three mechanisms are coupled: local geometric-semantic primitives define the measurable state, unsupervised clustering defines auditable analysis windows, and graph spectra with persistence define topology-sensitive fingerprints inside each window. We formalize hard feasibility gates, primitive embeddings, local assembly graphs, graph Laplacian spectra, persistent connectivity, and posterior semantic naming. For the pipe subproblem, we give a concrete port-graph Laplacian and prove that the multiplicity of its zero eigenvalue equals the number of hard-feasible connected pipe components. We then show how clustering and local spectra close the gap exposed by a negative case: ring-like geometry alone falsely confuses guardrail corners with handwheel-like structures. A real CAD scene with 7880 primitives is used as a numerical audit: all parsed primitives are embedded by type, size, pose, normalized position, local density, and neighborhood type histograms; Ward clustering assigns them to 14 structure clusters; the complete assignment is written back to a colored PHG file for visual inspection. The paper therefore provides both theoretical closure, through graph-spectrum invariants and hard feasibility, and numerical closure, through reproducible clustering, scene-level coverage, negative controls, and explicit claim boundaries</p>