The Cosine Drainage Theorem. A Rank-One Proof and a a Cross-Family Conjecture for Embedded Non-Regular Graphs

# The Cosine Drainage Theorem: A Rank-One Proof and a Cross-Family Conjecture for Embedded Non-Regular Graphs ## Author Michael Godat, Independent Researcher, Steilacoom, Washington, USA ## Abstract We study cosine-greedy traversal on high-dimensional embedded graphs and identify a systematic failure mode: progressive degree drainage, in which cosine-to-target selection descends the degree hierarchy at each step, converging toward a structural periphery where traversal budget is exhausted before reaching cross-domain targets. The mechanism is the directional inverse of the friendship paradox (Feld 1991): where uniform edge sampling biases toward hubs, cosine-directed edge sampling biases away from hubs. This occurs because trained embedding models place high-degree (general) concepts angularly near the distributional mean, while specific targets deviate from that mean. Cosine similarity, being purely angular and norm-invariant, systematically selects away from the mean direction and therefore away from hubs. We prove a Rank-One Cosine Drainage Theorem under the Chung-Lu generative model with two assumptions: (A3a) degree-mean angular alignment (β > 0), and (A3b) target-relative hub disadvantage. We show that β > 0 alone is not sufficient — artificially injecting angular-degree coupling via a non-mean direction produces anti-drainage (Experiment 31, ρ(β,T3) = +0.73). The operative geometry requires mean-directed coupling specifically. We derive miss probability bounds via a scissors effect (declining degree × accumulating frontier), locate the phase transition at d_drain ≈ 3.5 on a 39,220-node biomedical concept graph, and prove that waypoint injection eliminates drainage by construction (100% intermediate concept recovery vs 13.3% for endpoint-only strategies, McNemar p = 9.5 × 10⁻⁷). Multi-Anchor (dual-frontier) expansion outperforms single-source cosine on all 15 evaluated graph families — including scale-free, heterogeneous, co-authorship, co-purchase, citation, web-hyperlink, random (Erdős-Rényi), small-world (Watts-Strogatz), block-model (SBM), preferential-attachment (Barabási-Albert), and synthetic kNN — spanning drainage, weak-drainage, and anti-drainage regimes. The intervention is robust regardless of the local mechanism. ## Contents of This Upload 1. `cosine_drainage_theorem.md` — The formal theorem (frozen 17 August 2026) 2. `vgsg_thesis_paper.md` — Full thesis paper with all experimental results 3. `vgsg_conjecture.md` — The VGSG conjecture document 4. `retraction_orc_proxy.md` — Transparent retraction of ORC proxy mechanism 5. `validation_data_requirements.md` — What data is needed to reproduce 6. `README_ZENODO.md` — This file ## Code and Data Repository All experiment code (31 experiments), results (JSON), proofs, and gap-closure documents are available at: https://github.com/mjgodat/cosine-drainage-theorem The repository includes: - 31 experiment scripts (Python, PyTorch, GPU-accelerated) - All result JSON files - 6 proof documents - 5 gap-closure analyses - Reproducibility manifest ## Key Results | Finding | Evidence | |---------|----------| | Progressive drainage confirmed | 10/15 graph families, T3 ρ = -0.30 to -0.87 | | Multi-Anchor dominates | 15/15 graph families at matched budget | | Waypoint injection: 100% vs 13.3% | McNemar p = 9.5 × 10⁻⁷, budget 0.57x | | Predictive model R² = 0.92 | 3 features (Γ, log H, degree CV), LOO ρ = 0.924 | | Encoder-agnostic | nomic 768D + BGE 384D, same results | | K_mom = -1/2 theorem | Formally proved, verified across 7 distributions | | SBM anti-drainage boundary | T3 = +0.52, identifies conjecture boundary | | β > 0 insufficient for drainage | Exp 31: ρ(β,T3) = +0.73 with arbitrary direction | ## Experimental Scope - 31 experiments executed - 15 graph families tested (6 real-world, 9 synthetic/generative) - 2 neural encoders (nomic-embed-text 768D, BGE-small-en-v1.5 384D) - 15 validated hypothesis traces across 2 independent sets - 6 traversal policies benchmarked - All experiments on consumer hardware (RTX 4070, i7, 32GB RAM) ## License MIT License. See the GitHub repository for full license text. ## Citation ```bibtex @misc{godat2026drainage, author = {Godat, Michael}, title = {The Cosine Drainage Theorem: A Rank-One Proof and a Cross-Family Conjecture for Embedded Non-Regular Graphs}, year = {2026}, doi = {PENDING}, url = {https://github.com/mjgodat/cosine-drainage-theorem} } ``` ## Acknowledgments Developed through iterative research across multiple AI systems (Claude, Gemini, Perplexity, Grok) serving as research assistants, reviewers, and adversarial testers. This work emerged from building NeuroCrystal/PRSM, a geometric concept-traversal system for biomedical literature-based discovery. The trapping problem was encountered first. The architectural solution (waypoint injection) was built second. The theorem came last.