Can a 60 major+ player, play on a 55 AAA team?
No, according to the rules.
RI has 60M+ = 7 and 55AAA = 6 so technically you could but would count as a chip. That said, you should def call the Home Office and ask a ratings person to be sure, they will know you complete history and give you the correct information. Good luck!
The prohibition against Major+ also playing AAA applies only in the same age division ... So, the answer is: YES, according to the rules, if the player is otherwise eligible as to residency AND the target team has one of its three allowable "out-of-rating" exemptions available ...
