Hodge-projected echo-state networks with topologically anchored memory for chaotic flows

We introduce CHORD-ESN, an echo-state network that builds long memory from topology rather than from near-unstable tuning. The reservoir state lives on a simplicial complex as node potentials (0-forms), edge fluxes (1-forms), and face circulations (2-forms), and cross-degree interactions follow the laws of exterior calculus. A Hodge projection splits edge flows into exact, coexact, and harmonic components, and we assign a tiny leak only to the harmonic part. This yields a topology-anchored slow channel — with capacity set by the number of cycles — while standard components are damped by nonexpansive heat smoothing. We give a simple, verifiable echo-state (stability) condition via a small block-contraction bound, and the whole update uses sparse operators with intermittent lightweight solves. On chaotic and real-world benchmarks, CHORD-ESN improves long-horizon forecasting and attractor fidelity, and ablations that remove cycles or disable the Hodge split eliminate these gains. In short: cycles remember; CHORD-ESN makes that memory explicit, controllable, and provably stable.