Common-factor cancellation defect in a multistep recurrence

Canceling a common factor of the characteristic polynomials changes the allowed starting relations and can remove a parasitic mode rather than prove stability of the original recurrence. For example $\rho =(\zeta -1{)}^2$ and $\sigma =({\zeta }^2-1)/2$ fail zero-stability. On $y^{\prime }=0$, the original recurrence admits $y_n=y_0+n(y_1-y_0)$; starting perturbations $y_1-y_0=\sqrt{h}$ vanish but give divergent error at fixed positive time. The canceled one-step recurrence excludes these data, so its stability cannot be assigned to the original scheme.