The ArrowSpace Algorithm: From Graph Wiring to τ-Mode Spectral Search

ArrowSpace is a graph-native indexing engine that replaces (or augments) purely metric vector search with a spectral view of a dataset. Rather than building a graph over items, ArrowSpace transposes the data matrix into feature space, builds a sparse k-nearest-neighbour Laplacian over features (“graph wiring”), and attaches a single scalar — the synthetic τ-mode index λ — to every item by evaluating a bounded Rayleigh-quotient functional on that Laplacian. The resulting index supports λ-aware nearest-neighbour search, clustering, classification, compression and mechanistic analysis, depending on which Index Score Phase is plugged into the pipeline. This report documents the full algorithmic pipeline: graph wiring, clustering/sampling/compression, FastPair-accelerated Laplacian construction, τ-mode synthesis, λ-aware search, and the test-driven experimental evidence supporting each stage. τ-mode synthetic score, in particular, is an effective encoding for position and connectivity of a node into a single scalar value .