Stationary versus causal solution of a two-sided AR(1) equation ($|\varphi |\ne 1\text{ stationary};|\varphi |<1\text{ causal}$)

With nondegenerate white noise, a two-sided AR(1) equation has a unique weakly stationary solution exactly when $|\varphi |\ne 1$. The solution is causal for $|\varphi |<1$ and anticausal for $|\varphi |>1$. At $\varphi =\pm 1$, an n-term noise sum has variance $n{\sigma }^2$, while its difference-of-stationary-values representation has variance at most four times the stationary variance, a contradiction.