[F]or the word "soul" I'd use Wikipedia's basic definition: The soul in many religions, philosophical and mythological traditions, is the incorporeal and immortal essence of a living being.
That's not a definition that's useful here. When there are several contradictory and disparate abstract concepts behind a word, lexicographers must address them all. That is not a basis for testing the actual existence of any of the concepts. Regardless what word you assign to the entity having the properties you seek, testing the existence of that entity requires the properties to be definable.
You need to decide what object, entity, or force exhibits the properties that affect your conclusion and devise a way to quantify those properties such that they can be manipulated by a quantitative model. Otherwise you're just begging the question irrespective of lexical mechanics.
But, keep in mind that I don't use that word as it automatically accepts immortality, and I'd be begging the question if I used it.
You don't seem to have a problem begging the question by assigning arbitrary numbers to probabilities in your statistical model.
Here, I'm just talking about the "self," or individual consciousness and trying to show that H -- the scientific expectation that each (potential) self has one, finite, life (at most) -- is misled.
But that's not what you set out to prove. You said -- as I quoted above -- that you would prove immortality by means of mathematics. Now you're desperately trying to change horses and reverse the burden of proof. So if you intend now to embark upon a new quest, please provide closure for your first proposition. Do you concede that you are unable to prove immortality mathematically?
There is no "scientific expectation" at work here, but thanks for letting the veil slip ever so slightly on your crusade against "science."
The proposition that life is as we observe it to be is simply the logical conclusion based on evidence. You are the one saying that there is more to life than what we can observe. In that formulation, the "scientific expectation" of one finite life is simply the null hypothesis. And yes, you have the burden to falsify the null. Your critics don't have a burden to uphold it; that's what it means to be a null hypothesis.
But you don't get to falsify the null by handwaving vaguely toward a grab-bag of dissimilar and contradictory propositions whose only common element is denial of the null. You especially don't get to do that in a statistical context, where specific propositions have numerical combinatorics and severability. Your frustration along these lines is evident and understandable, but you don't get to alleviate it by begging your critics simply to relax their standards until your claim survives.
Think about interpreting your frustration as the failure of your claim to survive rational scrutiny.