<p>Point cloud registration (PCR) remains challenging in low-overlap, cluttered scenes where limited co-visible geometry and ambiguous features undermine correspondence reliability and geometric consistency. To address this challenge, we propose a novel end-to-end trainable coarse-to-fine registration framework, HGraphDyFusion. The coarse stage first extracts geometry-aware features with attention and produces initial point correspondences. We treat each correspondence as a node and construct a hypergraph over the correspondence set. We iteratively propagate high-order relations on the hypergraph to suppress outliers and keep consistent matches, yielding a stable and reliable set of correspondences. These correspondences then guide a Fourier-encoded fine matching module for precise alignment. At the coarse level, Dynamic Hypergraph Consistency Modeling (D-HCM) retains the backbone pair descriptors and augments them with two complementary cues in the form of a difference term and a similarity term. Under an incidence mask, circle (node-hyperedge-node) attention between nodes and hyperedges estimates hyperedge confidences. A confidence-guided reweighting and pruning step converts the initially dense hypergraph into a clean inlier subgraph. At the fine level, Fourier Positional FiLM (FP-FiLM) encodes anchor-relative offsets. The offsets are normalized by the neighborhood radius and embedded with multi-frequency Fourier features. This high-frequency encoding captures subtle local geometric variations and boosts fine-grained point discriminability for correspondence refinement. This improves discrimination on smooth and repetitive structures and yields accurate point-wise correspondences in low-overlap scenes. Extensive experiments on 3DMatch, 3DLoMatch, and KITTI show consistent gains under noise, occlusion, and low overlap. Under the most challenging low-overlap regime, registration recall increases by up to 0.8 percentage points.</p>