Period Snapshots

Two-Cycle Must Return and Not Stop

A two-cycle is a pair of states that swap under the map and return after two steps, f(f(x)) = x with f(x) != x, so neither state alone is the resting point. This verdict is exact for the stated map, parameters, and iteration count over a finite snapshot. Real systems add transient behavior, noise, parameter uncertainty, sampling limits, and model error that can mimic or hide a true cycle.

A period-two witness needs both a return and a non-fixed step. 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 two-cycle must return

The reflection map sends one quarter to three quarters and then back to one quarter.

14→34→14\frac{1}{4}\to\frac{3}{4}\to\frac{1}{4}
Two-cycle returnThe return value is checked by two updates.mapId=reflectionparameter=1xValue=1/4yValue=3/4returnValue=1/4returnedBit=1 bitnonfixedBit=1 bitperiodTwoBit=1 bit

It must also not stop after one step

Because three quarters is not one quarter, the period-two bit is 1.

f(x)≠x and f(f(x))=x⇒1f(x)\ne x\ \text{and}\ f(f(x))=x\Rightarrow 1
Non-fixed period witnessA fixed point cannot spoof a two-cycle.mapId=reflectionparameter=1xValue=1/4yValue=3/4returnValue=1/4returnedBit=1 bitnonfixedBit=1 bitperiodTwoBit=1 bit