A structured understanding of intraoral 3D scans is essential for digital orthodontics. However, existing deep-learning approaches rely heavily on modality-specific training, large annotated datasets, and controlled scanning conditions, which limit generalization across devices and hinder deployment in real clinical workflows. Moreover, raw intraoral meshes exhibit substantial variation in arch pose, incomplete geometry caused by occlusion or tooth contact, and a lack of texture cues, making unified semantic interpretation highly challenging. To address these limitations, we propose ArchMap, a training-free and knowledge-guided framework for robust structured dental understanding. ArchMap first introduces a geometry-aware arch-flattening module that standardizes raw 3D meshes into spatially aligned, continuity-preserving multi-view projections. We then construct a Dental Knowledge Base (DKB) encoding hierarchical tooth ontology, dentition-stage policies, and clinical semantics to constrain the symbolic reasoning space. We validate ArchMap on 1060 pre-/post-orthodontic cases, demonstrating robust performance in tooth counting, anatomical partitioning, dentition-stage classification, and the identification of clinical conditions such as crowding, missing teeth, prosthetics, and caries. Compared with supervised pipelines and prompted VLM baselines, ArchMap achieves higher accuracy, reduced semantic drift, and superior stability under sparse or artifact-prone conditions. As a fully training-free system, ArchMap demonstrates that combining geometric normalization with ontology-guided multimodal reasoning offers a practical and scalable solution for the structured analysis of 3D intraoral scans in modern digital orthodontics.