= 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 $y'=0$, 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.
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