A piecewise map picks a different formula depending on where the current state sits, so each ledger row must record the branch that fired alongside the value. It models discrete switching behavior, not smooth continuous dynamics. The exact results hold only for the stated branch rules and the given starting input.
A piecewise map is honest only when the branch choices are checked. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
A branch choice is part of the source
Starting at three quarters uses the high branch and lands at one half.
2(1−43)=21
The equality boundary is explicit
At one half the low-or-equal branch gives 1, then the high branch gives 0.