Faster, Fewer, & Farther: Granularity-Aligned Hierarchical LiDAR Relocalization in a Shared Spectral Geometry

Global LiDAR relocalization requires transferable association across growing maps, where geographically distant places can share similar structure. Exhaustive flat frame retrieval preserves a simple comparison interface but incurs map-linear search and increasing exposure to distant aliases. Hierarchical retrieval reduces this search, yet makes coarse routing an irreversible geographic hypothesis-pruning operation: a discarded location cannot be recovered by fine matching. We identify spatial, granularity, and embedding-consistency mismatches: coarse candidates can be appearance-coherent yet geographically distant, routing may compare evidence at unequal granularities, and separately fitted stage representations may disagree under domain shift. F3Loc aligns support and granularity in a simple hierarchy with no routing-specific representation learning. Query context routes to spatially coherent map regions, after which the current frame localizes among the retained frames. The proposed shared-basis spectral calibration establishes a fixed comparison geometry before aggregation, and granularity-aligned routing preserves the resulting frame-neighborhood semantics from regional routing to final localization. Across Oxford, NCLT, and MCD, F3Loc discards 96.1–99.5% of map frames, yielding 2.35–8.90× speedups over same-encoder exhaustive retrieval. Mean translation error falls by 54–56% on Oxford and 31–52% on NCLT and MCD. Under the same no-target-training protocol, F3Loc achieves substantially lower localization error than industrial ANN and hierarchical retrieval controls, while attaining translation accuracy competitive with, and on several routes better than, in-domain regressors. Code will be released.