I have been on this site for a number of years and I still have no idea what The SAI is.... Does anyone one know?

It doesn't do anything. I think SAI is the Subetan equivalent of the CIA/FBI. I guess stuff was planned for it but it never got expanded upon. :)

It was a brief location of a restock war in which you were rewarded for donating "sun" items, and it was the location of the pastry box police during the Fantine/Skitters/Ecks event.

Yeah, it's like a secret government agency. It's been used in a couple of events before, but not a whole lot. Maybe it will be incorporated into something new in the future, who knows.