arXiv:2610.34109v1 Announce Type: new Abstract: Geo-filtered approximate nearest-neighbor (GFANN) search retrieves the objects inside a query-specified geographic disk whose feature vectors are closest to a query vector, a core operation in location-aware Web search. A disk filter on a standard proximity graph invalidates out-of-disk routing vertices; on selective disks, these removals break greedy paths and recall drops sharply. Existing methods scan all in-disk objects, over-fetch unfiltered candidates, or maintain separate sub-indexes across spatial partitions, trading latency for index overhead. We propose REACH, a single-graph index based on row-local active-order invariance: if every vertex stores its outgoing neighbors in a fixed-width row sorted by vector distance, any disk filter masks out-of-disk entries while surviving in-disk (active) neighbors remain vector-sorted. Offline, each vertex merges globally closest vector candidates with a nearest-first geographic stream into one row; online, beam search evaluates vector distances only for the first l active entries of each visited row. Because co-located objects share the same distance to every query center, every disk admits a coordinate group in full or not at all, decomposing active connectivity into intra-group and inter-coordinate conditions. Under a spatial marked-site model with repeated coordinates, we prove that polylogarithmic out-degree guarantees strong connectivity across all disks and constant-factor distance reduction per hop on compact disks, while a cut-rank threshold characterizes the minimum scan budget l. Across four datasets up to 10^7 objects, one REACH index per dataset outperforms SOTA methods, achieving 2.49x higher geometric-mean QPS than the strongest baseline at 0.95 Recall@10 on selective queries, while building 3.8x-11.9x faster and using 37%-45% less space than a SOTA spatial-graph index.