Sub-Laplacian Spectral Hashing for Topological Proximity Indexing - GraphRAG

Description Modern semantic search depends on expensive distance computations over massive embedding indices, where this single operation can dominate inference cost. This work poses a deliberately unconventional question: can such an index be queried using only its connectivity—its raw topology—after every coordinate vector has been permanently deleted? The success is that each node is assigned a compact binary identifier built by recursively bisecting the graph along its spectral structure. Nodes sharing a prefix naturally fall within the same region of the network. As a result, candidate retrieval reduces to a single bitwise (XOR) operation per node—no distances are computed during pruning, and no embeddings are stored. The central contribution is not only computational but predictive. Whether the method yields a strong signal—or none at all—is governed by a single classical graph statistic that can be computed in seconds. This: Refutes the assumption that well-defined communities are required for the method to work. Challenges the belief that high-dimensional embeddings inherently degrade performance. Provides a prior diagnostic rule: one number determines in advance whether a network is richly indexable or topologically silent. We trace this behavior from idealized lattice structures, through varying embedding dimensionalities, to a real-world collaboration network indexed in sub-second time on mobile hardware, without GPU acceleration. The result is a storage-free, constant-time routing layer for graph databases, together with a practical criterion to predict its effectiveness before deployment.