Persistence changes character as an autoregressive coefficient approaches one: for each fixed 0<a<1, survival above zero decays exponentially, whereas at the unit root symmetric random-walk survival is of order n−1/2. We study this transition for AR(1) sequences driven by symmetric α
Nearby in the stack
-stable innovations and write
Λ(a,α)
for their exponential persistence rate. The entire chain admits an exact representation through a single stable Lévy process observed on a geometrically expanding time grid. Comparison with continuous half-line survival gives
Λ(a,α)≤2αlog(1/a)
. For
0<α<2
, this bound disproves the stable specialization of a conjecture of Hinrichs, Kolb and Wachtel for regularly varying innovation tails. To obtain a lower bound of the same near-unit order, we combine stable closure under subsampling with a monotonicity coupling. This proves
Λ(a,α)≍log(1/a)
as
a↑1
and shows that the ratio
Λ(a,α)/log(1/a)
converges to a limit in
(0,α/2]
, equal to its supremum over
0<a<1
. Finally, a Lamperti transformation reduces identification of this constant to a dense-sampling persistence problem for a stationary stable Ornstein--Uhlenbeck process. Existing Gaussian theory determines the sharp value at
α=2
. For
0<α<2
, identifying the value requires controlling paths that cross below zero and return above zero between consecutive observations.