A reflection-parameter-period cross-scan varies the map parameter across rows and reads off the orbit's period, showing where fixed points give way to cycles as the parameter changes. The arithmetic is exact only for the stated parameters, periods, and rows in the ideal discrete-map model. Real systems add continuous-time dynamics, noise, parameter drift, finite precision, model mismatch, sampling choices, and measurement uncertainty.
A reflection map family is scanned with finite period-row tests. 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 reflection-family row is a finite two-step test
The candidate is fixed at one quarter. Each parameter row checks the first update and the return update.
fp(x)=p−x
Period labels follow the two checked updates
The first parameter row is period one. The next two rows return after two steps without stopping after one step.