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There are Four types/applications of Type O carriers. They are based on what gears are going onto them. 3.23:1 and 3.42:1 fit DIFFERENT carriers, as shown here.
Alain, do you actually have a 588 carrier? Or OEM 3.42:1 gear set?
On the aftermarket Richmond/Yukon, etc., 3.42:1 gear set, the ring gear is designed to fit the 671 carrier.
Just trying to help cut through the fog, as I was looking for literally years for an OEM 3.42:1 gear set to go on my NOS 588 carrier. I had a Richmond aftermarket set on hand. I eventually came across a 671 carrier that I bought to mate up with my Richmond gear set.