ACCORDING TO HER, the entire array is synchronized.

True, whether he exists or not, she concludes that her array of clocks are synchronized.

The established fact is that she concludes that her array of clocks are synchronized. That doesn't require that anyone but her (and her perpetually-inertial helpers permanently co-located with those clocks) has to agree that her clocks are synchronized. Other perpetually-inertial observers who are moving with respect to her don't agree that her clocks are synchronized. And the accelerating observer, or an observer who has accelerated recently, doesn't necessarily agree with her either, even if he is stationary with respect to her for some period of time.

I said, "When she is 40 years old, her entire array of clocks displays 40,

**simultaneously in that reference frame**," and you were correct to agree. In SR simultaneity depends on the reference frame, not on individual people. That way, we can unambiguously say that the event of the traveler jumping off the train has these coordinates in the reference frame of the train tracks: (x, t) = (34.64, 40.00) and these coordinates in the reference frame of the train: (x', t') = (0.00, 20.00) and that one can use the Lorentz Transforms to get either one from the other.

If you were correct in your assertion above, (which you are not), then the event of the traveler jumping off the train would also have these coordinates in the reference frame of the train tracks: (x, t) = (34.64, 10.00) but only according to one guy in that frame, only for the traveling twin! Notice that you cannot even use the Lorentz Transforms to get those coordinates, because it is not SR, you made it up.

Also, how could the traveler have gone from x=0.00 to x=34.64 in only 10.00 years time? He would have to travel over 3 times the speed of light. It's not physics anymore, its nonsense. There's a reason we call the time t=40.00 in that frame, because that is how long it takes for the traveling twin to reach x=34.64 when traveling at v=0.866c there is no disputing it. The jump-off-the train event has nothing to do with the time t=10.00, and to claim it does is simply WRONG.*

*But it is true that the traveler was correct in assessing that her age was 10 at the time he jumped off the train, but only because he was on the train then, so her clock array was not synchronised then. Once he is stationary on the ground he can no longer say that her clock array is not synched.