As-built BIM reconstruction from point clouds demands semantic, volumetric, parametric, and editable building components, beyond 2D layouts or planar abstractions. Yet existing scan-to-BIM pipelines often rely on handcrafted rules, heuristic optimization, or manual correction, while learning-based methods fall short of recovering wall thickness, opening geometry, and component-level BIM structure. We introduce CRAB, a BIM-ready cuboid reconstruction framework that encodes walls, doors, windows, beams, and columns as compact architectural cuboids with IFC-compatible semantics and parameters. CRAB bridges point clouds and editable BIM by combining ProposalNet, which predicts coarse component cuboids, with AssemblyNet, a relation-aware Flow Matching model that refines geometry, predicts structural relationships, and enables automatic structural correction and human-guided editing. To address limited cuboid-level supervision, we extend Structured3D with component-wise cuboid annotations and construct cuboid-based procedural scenes. Experiments show CRAB surpasses floor plan and plane-based alternatives in component completeness, geometric accuracy, and generalization. For interior wall thickness estimation, CRAB achieves a mean error of 2.80 cm, with 79.50% of matched walls within 3 cm. The project page is available at https://vcc.tech/research/2026/CRAB.

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