The stochastic web map — a kicked oscillator generating a web-like chaotic phase space — has a symmetry parameter q that controls the geometry of the web (triangular, square, hexagonal) and a nonlinearity parameter K that controls how wide the chaotic channels are. Different q values produce visually distinct structures. The escape dynamics — how a particle starting inside the web eventually escapes past a boundary — ought to depend on both parameters, since the geometry determines which paths are available.
It doesn't depend on q (arXiv:2603.20888). When the survival probability and escape frequency are measured and time is rescaled by the characteristic timescale n_typ ∝ K⁻²h², where h is the escape horizon size, the statistics collapse onto a single curve regardless of the web's symmetry. Triangular, square, and hexagonal webs produce the same escape statistics. The geometry is irrelevant; only the channel width and horizon size matter.
The characteristic timescale captures the competition between diffusion rate (controlled by K — wider channels mean faster transport) and the distance to the boundary (h). The escape is a transport problem, and transport through the web depends on how easily a particle moves through the channels, not on how the channels are arranged.
Deviations appear at large K and small h, where the quasilinear approximation (which predicts the K⁻² scaling) breaks down. But within the regime where transport is diffusive, the universality holds.
The structural insight: the web's symmetry determines the local structure — which directions the channels run, how they intersect, what the unit cell looks like. But escape is a global property: how fast does transport carry a particle across the entire web? The global transport averages over the local geometry. The symmetry parameter sculpts the details; the transport parameter controls the outcome. When you care about the outcome (escape), the details (geometry) become a rescaling, not a variable.