No, we won't.
In order for any thinking to be done at the fringes, the fringes have to
be moving slower than light. (See "Universal Non-Point of Light.") To
put it another way, you can't lose touch, in the warm n' fuzzy sense,
with something that isn't thinking. So fringe A can send a beam of
light to fringe B, going in the opposite direction, which will reach it
in a finite amount of time. Fringe B's reply will reach fringe A, again
in a finite amount of time (*). So the fringes never fall out of
communication - as long as relativity applies, we remain one big happy
geometrically expanding family.
(*) In fact, exactly the same amount of time. If B measures the amount
of time it takes the ray to reach A, and the amount of time it takes the
reply to return (assuming composition is instantaneous), they are
exactly equal - assuming B's velocity is constant. Question 1: Why?
Doesn't B move farther away from A after the reply is sent? Question
2: B says sending=reply. Fringe A says the same thing, except A's
"sending" is B's "reply", and vice versa. Doesn't this mean
sending=reply=sending=reply=sending..., thus making (by induction)
communication time constant even as A and B move farther and farther
away from each other?
--
sentience@pobox.com Eliezer S. Yudkowsky
http://tezcat.com/~eliezer/singularity.html
http://tezcat.com/~eliezer/algernon.html
Disclaimer: Unless otherwise specified, I'm not telling you
everything I think I know.