Period Snapshots

Two-Cycle Must Return and Not Stop

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