Whilst I understood the reason for the splitting in the characters memories I didn't really buy why the splitting would have been rectified when they both were reunited. Is it just the simple fact that their memory of each other on that previous time line was confirmed?
As for whether or not it was the same Jake... the fact of the paradox suggests it was. However ....
later books spoilers...
Spoiler:
Wolves of the Calla suggests that they are seperate Jakes.
That being said, it could be explained to some extent by the whole quantum universe theory where seperate universes are spawned considering what decision is made. Therefore the two Jakes are now seperate, one alive, one dead, but before Roland saved Jake by killing Mort they were the same. Of course I'm not sure why the save Jake would have the memories of the other (regardless of whether or not they are the same, since he was saved before he met Roland) but that could be put down to Ka-tet and his power in the touch coming into play.
That makes some sense but what I am hearing is that there is no body of a young boy under the mountain?
Somehow at the bottom of the fall he returned? Or it never happened--I'm not sure.
However, on more reflection of the original question (because this thing about Jake poses all sorts), I do seem to remember that he "relived" it somehow. And I also seem to remember a place where he tells Roland..."you dropped me" right after they are reunited in this book.
I'm not totally sure how it could work either but I think the question about Jakes actual death under the mountain is somehow key.
But when Roland goes through the third door, he saves Jake from being thrown in front of the Cadillac (in some way--we know that there was issues about it not being the "right" day, but we know he didn't die under the wheels of a car).
If he doesn't die in the streets of New York, he can't go to the Way Station, and Roland can't drop him. That's where the duality that causes Jake's insanity occurs. He remembers dying in New York and remembers all the events afterward, but his brain is arguing with him. Even in his "My Understanding of the Truth" poem, he says that Roland dropped him.
I believe that he did die when Roland dropped him...
Having just reread "The Drawing of the Three" and "The Wastelands" it’s pretty clear that it is the same Jake, it’s not really given any room for the possibility that it isn’t. And Jake (just like Roland) remembers everything from both versions of the timeline. The entire reason Roland and Jake are going insane in book 3 is that Roland saved his Jake in book 2 which is the same Jake as in book 1 and 3. There wouldn’t be a paradox otherwise.
Yeah, this episode seems clear enough, given King's right to fictional license. Both Schrödinger's Gunslinger and his boy have what amount to split personalities, which King certainly spends enough time describing, presumably brought on by the original incident in The Gunslinger when Roland abandons Jake. It's driving them nuts, and they are both cured when Jake is pulled through the door into Roland's world. But nothing it said about memory, so we may believe that Jake remembers the events of both tracks of the "split" period.