If the slope magnitude at the fixed point exceeds one, |f'(x*)| > 1, small deviations grow each step, so the point is unstable and a stability label is rejected. The test turns on the local linear slope, exactly as in the stable case but flipped. It is exact only for the stated map and linearization; real systems have measurement noise and model mismatch, so borderline slopes are decided with uncertainty.
A valid fixed point can have a rejected local stability bit. 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 fixed point can still fail the stability contract
The repeller map also has candidate one half as a fixed point.
g(21)=23⋅21−41=21
Slope greater than one is rejected
The exact local slope is three halves, so accepted bit is 0.