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In this case, rebirth is precisely the same as reincarnation.

Yes. And we see folks like Brahmavamso who uses the two terms interchangeably, for example, on his YouTube video on the subject ( The Process of Rebirth ). The tibetan religions don't bother to try to draw a distinction between the two, and simply use the word "reincarnation". At least they are honest with themselves in that respect.

The motion of the blue field is entirely determined by the motion of the yellow field, but is separate and distinct from the yellow field.

The motion of each ball in either field is predicated upon a measurable, falsifiable mechanism: the physical contact of another ball. Made-up theories about reincarnation/"re-birth" abound, and none of them bothers to suggest a plausible mechanism for transference of "whatever-it-is-that-is-transferred" -- which cannot help but be an "atta', no matter what kind of logical gyrations one employs in their made-up "explanation", which cannot help but be anything more than a fable.

Of course, as soon as that fable is turned into a belief or a belief system, a Flat-Earth Big Lie is born -- and as soon as it is attached to an ethical system, a Flat-Earth-Big-Lie Religion is born.