A stability-margin-boundary scan measures how far each fixed point's slope sits from the |f'(x*)| = 1 threshold across rows, marking the margin that separates stable from unstable. The verdict is exact only for the stated slopes, fixed points, and rows in the ideal linearized discrete-map model. Real systems add continuous-time dynamics, noise, parameter drift, finite precision, model mismatch, sampling choices, and measurement uncertainty.
The strict stability gate is scanned through positive, zero, and negative margins. 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 zero slope sits inside the stability gate
The logistic fixed point at one half has slope zero, leaving a full margin below one.
m=0,1−∣m∣=1
Magnitude one is the rejected edge
The reflection row returns to itself but its slope magnitude is exactly one, so the strict gate rejects it.
m=−1,1−∣m∣=0
A larger slope gives a negative margin
The repeller row shows the outside side of the same gate.