The Cosine Drainage Theorem the Cosine Drainage Theorem: A Rank-one Proof and a Cross-family Conjecture for Embedded Non-regular Graphs

Source abstract

Cosine-similarity-biased traversal of high-dimensional embedded graphs systematically fails to reach graphreachable targets under finite budgets-a phenomenon termed the Vector-Graph Semantic Gap (VGSG). This paper identifies the mechanism as progressive degree drainage: in embedded graphs where cosine-to-target selection systematically underweights higher-degree frontier candidates, greedy traversal exhibits monotone degree regression toward a lower effective degree regime. The Rank-One Cosine Drainage Theorem proves this mechanism under the Chung-Lu generative model with measured angular-degree coupling β > 0. The broader Cosine Drainage Conjecture-that the same mechanism operates on all non-regular embedded graphs satisfying the coupling condition-is supported by empirical evidence across 15 graph families (degree CV 0.10 to 2.69) and two independent neural encoders (nomic-embed-text 768D, BGE-small-en-v1.5 384D). The theorem derives the miss probability as a product of per-step alignment factors governed by a scissors effect (declining degree versus accumulating frontier), locates the drainage-driven phase transition, and proves that multi-anchor and waypoint-injection architectures bound the failure mode. At matched budgets, Multi-Anchor expansion outperformed single-source cosine in all 15 evaluated graph families, including graphs exhibiting drainage, weak drainage, and anti-drainage. Every primary quantity is computable from the graph's degree distribution and embedding angular statistics, rendering predictions falsifiable on any new graph-embedding pair.