A map with saturating feedback bends its response so that beyond a threshold the output stops tracking the input, and the branch test checks whether that cap holds for the stated parameter values. The check is exact only for the stated map rule, branch, iteration points, and parameters in the ideal discrete-map model. Real systems add continuous dynamics, noise, unmodeled feedback, finite sampling, parameter uncertainty, and source-traceability limits.
A clipped feedback recurrence validates each branch and clamp. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Feedback follows the selected branch
The saturating gain sends one quarter to one half and then to 1.