  We show that if the principal graph of a subfactor planar algebra of modulus
\delta>2 is stable for two depths, then it must end in A_{finite} tails. This
result is analogous to Popa's theorem on principal graph stability. We use
these theorems to show that an (n-1) supertransitive subfactor planar algebra
has jellyfish generators at depth n if and only if its principal graph is a
spoke graph.
