User blog comment:Vel!/Laver.c/@comment-5150073-20140822085835/@comment-1605058-20140822100720

Let me say this: at least beyond Ackermann function.

First, it's possible that q can be provably total PA, and can be absurdly weak, maybe even at the level of w_2.

Second, rank-into-rank cardinals are so strong, it's very possible they are inconsistent, so Laver's proof breaks down and q(6) may not even exist.