Start anywhere and keep applying the same rule — press cosine again and again — and every starting point is drawn to the same still value it can never leave. That is the Return: the many turning back to the One. Slide the iterations (or tap) and watch the paths converge.
A Banach fixed-point iteration on f(x)=cos(x). The map is a contraction on [0,1] (|f′|=|sin x|<1), so by the contraction mapping theorem every start converges to the unique fixed point x* ≈ 0.739085 where cos(x*)=x* — geometrically, the cobweb staircase between the curve y=cos(x) and the line y=x spirals into their crossing. The step |xₙ₊₁−xₙ| shrinks geometrically to zero. A fail-loud self-check throws unless the iteration converges to the cosine fixed point and the step size strictly decreases.
One contraction map (cosine) on a bounded interval; not every map has a single attracting fixed point (that is what the-logistic explores). The iteration, the fixed point and the shrinking step are computed exactly.