HELM - Topology-Adaptive Geometric Routing with Non-Equilibrium Path Estimation and Orthogonal Channel Encoding

Abstract

HELM (Hyperbolic Embedding with Langevin Manifolds) is a layered routing architecture for networks where topology and traffic both change faster than conventional protocols comfortably assume. It combines three separable layers: a geometric forwarding layer that selects between hyperbolic and E8-inspired embeddings based on the topology's degree distribution, trading path stretch for substantial routing-table compression; a Cayley-Dickson-inspired channel encoding layer that instantiates seven mutually orthogonal channels in routing metadata; and a Jarzynski-inspired path-cost estimator paired with Langevin-style perturbation for decision-making on non-stationary latency observations. This record contains the preprint describing the architecture and its prototype evaluation on synthetic 200-node topologies, including a routing-under-uncertainty study in which both HELM and a Dijkstra baseline observe only noisy link state. The work is presented as a research prototype and candidate architecture rather than a production-ready protocol, with maturity deliberately reported per layer: the geometric and estimation layers carry the strongest evidence, while the channel-encoding result is scoped to encoding space and does not claim end-to-end traffic non-interference in deployed systems. The intended near-term application is LEO satellite constellations, where inter-satellite links churn on 2–5 minute timescales and onboard state budgets are tight. Version 1.0.1 (2026-08-21) — corrections to the deposited manuscript. Version 1.0.1 (2026-08-21) — corrections to the deposited manuscript. No reported measurement changed except the Section IV-C path-stretch figure, which was corrected from 1.49x to 1.58x to match Table II andthe limitations section. The remaining changes add previously omitted uncertainty and correct descriptive text: - Table I: standard deviations added to the Relative Difference column; caption now states the ensemble structure (5 independent seeds x 100 trials, 200 latency samples per traffic model per trial).- Table V: standard deviations added to the Latency Improvement column; caption now states n = 50 and the trial structure, and notes that Table IV uses 60-node graphs, so the two are not directly comparable.- Section IV-C: path stretch corrected from 1.49x to 1.58x.- Section VIII-F: the evaluation is described as 5 independent seeds x 100 trials, correcting an earlier "30-trial" description that contradicted the Table IV caption.- Table of contents: section numbering corrected (XI Discussion, XII Discussion/Limitations, XIII Conclusion); Appendix A now precedes the references.- Author contact address corrected.