A reflection-return-gap-period scan varies the return gap across rows, showing how the map period closes only when the reflected iterate lands back on the expected cycle. The verdict is exact only for the stated parameters, return gaps, periods, iterates, and rows in the ideal discrete reflection-map model. Real systems add continuous-time dynamics, noise, parameter drift, finite precision, model mismatch, sampling choices, and measurement uncertainty.
Reflection rows separate fixed return from period-two return. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
The midpoint returns but is not period two
One half maps to itself. The return bit is not enough without a non-fixed first step.
x=y=r=21
One third gives a true two-step return
The first update moves to two thirds, then returns to one third.
x=31,y=32,r=x
A larger first-step gap is still period two
Changing the starting point changes the gap but not the exact two-step return contract.