The way US AMEX works in my experience on getting the rebate:
All charges from Thursday (week 1) through next week's Wednesday (week 2), the rebate will post on Week 3's Friday.
There seems to be a minimum 9 days (15 max) before the charge gets the rebate.
Example with Thursday being the 1st of the month:
Week 1: make charge on Thursday the 1st (or Fri the 2nd, or Sat the 3rd)
Week 3: rebate shows up Friday the 16th
Example with Wednesday being the 31st of the month:
Week 1: make charge on Wednesday the 31st (or Sun the 28th, or Mon the 29th, or Tues the 30th)
Week 2: rebate shows up Friday the 9th
This is how it currently works from my understanding. I've seen some rebates posted outside this timeline, but for the majority of rebates, this is how it works for me.
Charges on Wednesday is the cut off to get the rebate for next week's Friday.
Amex uses two different dates, the one you see next to the transaction and the date that the transaction actually shows up on your activity where you see it. That's why you see posted dates for the rebate and transaction within 1 day of each other, but it doesn't show up until it's fully processed on that Friday following the schedule. So just check on Fridays for the rebates.