Some systems have more than one resting state, and where you END depends entirely on where you START. Each stable state sits at the bottom of a BASIN of attraction; every starting point belongs to exactly one basin and rolls down into it. The dividing ridge is a knife-edge. It’s the same two-basin geometry the corpus found in a transformer’s own dynamics. Slide the start across the ridge and watch the fate flip.
An attractor is a state (fixed point, cycle, or strange attractor) toward which nearby trajectories converge; its BASIN OF ATTRACTION is the set of initial conditions that end there. For the bistable flow ẋ = x − x³ there are two stable attractors at x = ±1 and an unstable one at 0, so the basins are the positive and negative half-lines split by the ridge at 0 — the outcome is PATH-DEPENDENT on the start. Basins can be smooth or fractally interwoven (riddled). This mirrors the two-basin convergence the corpus measured inside a transformer. A fail-loud self-check throws unless positive starts reach +1 and negative starts reach −1. ◆ real dynamical systems, node-verified.
A clean 1-D bistable is shown (two smooth basins); real systems can have many attractors with fractal basin boundaries. The path-dependence and the ±1 attractors are exact. Kin to the corpus’s jacobi two-basins finding.