Geometric Commensurability, Density-Gated Saturation, and Frequency-Robust Recurrence in a Helical Borrowing-Flow Toy Model

We investigate a minimal one-dimensional helical borrowing-flow toy model to test whethergeometric commensurability and density-gated saturation can support persistent, recurrent, andscale-robust circulation structures in a driven–dissipative system. Vorticity serves as a proxy forloop strength, while density-dependent saturation and resonance selection regulateamplification and stability. By fixing the helical winding number and varying the domain length,we find that coherence, attractor persistence, and longer-lag recurrence are consistentlyenhanced when the helix length is commensurate with the underlying topology. In contrast,incommensurate geometries exhibit weaker peaks, faster damping, and degraded recurrence.Normalized circulation density remains approximately invariant across scale, indicating thatextension does not dilute dynamical intensity. We further explore an optional frequency-tunedregime in which reduction-like events occur at gamma-band rates without loss of coherence,demonstrating temporal robustness of the underlying geometry. These results suggest thatgeometric commensurability and density gating provide a general mechanism for stabilizingextended recurrent dynamics without fine tuning.