A value that maps to itself is a fixed point, not a genuine two-cycle, so the period-2 test must reject any candidate with f(x) = x. That distinction is exact for the stated map, parameters, and iteration count over a finite snapshot. Real data add transient behavior, noise, parameter uncertainty, sampling limits, and model error, which can make a fixed point look cyclic for a few steps.
A fixed point can return without being a two-cycle. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.