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We present CAGE (Continuity-Aware edGE) network, a robust framework for reconstructing vector floorplans directly from point-cloud density maps. Traditional corner-based polygon representations are highly sensitive to noise and incomplete observations, often resulting in fragmented or implausible layouts. Recent line grouping methods leverage structural cues to improve robustness but still struggle to recover fine geometric details. To address these limitations, we propose a native edge-centric formulation, modeling each wall segment as a directed, geometrically continuous edge. This representation enables inference of coherent floorplan structures, ensuring watertight, topologically valid room boundaries while improving robustness and reducing artifacts. Towards this design, we develop a dual-query transformer decoder that integrates perturbed and latent queries within a denoising framework, which not only stabilizes optimization but also accelerates convergence. Extensive experiments on Structured3D and SceneCAD show that CAGE achieves state-of-the-art performance, with F1 scores of 99.1% (rooms), 91.7% (corners), and 89.3% (angles). The method also demonstrates strong cross-dataset generalization, underscoring the efficacy of our architectural innovations. Code and pretrained models are available on our project page: https://github.com/ee-Liu/CAGE.git.