There’s quantity, and there’s representation.
Representations are constructions that you have to build up before the construction “equals” a quantity (that was conceptually “already there” and didn’t need building).
Building the representation of (0.\dot9) needs endlessness, so writing out the representative construction never ends (or “ends” with indefinite continuation implied) with the quantity that it does equal, which is 1.
Any “gap” is at most hinted at by the representation, not the quantity that it’s building up to, but even then it is only hinted at to the uninitiated who impatiently second guess what the representation would look like if it were to be completed at any given finite point(s) along the way - and they thus conclude that it’ll never get there and there will always be a gap.
You don’t go (\frac{circumference}{diameter}\neq3.141592…\neq\frac4{1}-\frac4{3}+\frac4{5}-\frac4{7}+\frac4{9}-\frac4{11}+…\neq4\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}) because there’s always got to be some “gap” at any conceptual “end” point to the representation implied by that contradiction in terms, that prevents it from ever equalling (\pi). No, they all equal the quantity “pi” because of the fact the construction contains some “incomplete” indetermination to it, not in spite of it.
A fraction with finite terms (C\div{d}) isn’t resolved, the explicit decimal places format isn’t finished, the infinite expansion of explicit fractions doesn’t terminate, the infinite series never gets to that “infinity” - the representation of (\pi) just pretends the irrational and transcendental quantity of pi is finite to make it easier and more compact to accept that all these incomplete constructions really do equal (\pi) in exactly the same way as (0.\dot9) really does equal (1).
Yet somehow, non-mathematicians will generally all accept any of these notations of (\pi) but not the notation of (1) as (0.\dot9). Forgive the pun, but this is entirely irrational.
I’m only able to make sense of some of what you say because the intelligence tests on which I score highly are mostly testing how well someone can extract patterns and sense from increasingly obscure sets of information. When Magnus says to me “It seems like you’re one of those rare people who can understand Ecmandu”, he seems to think that even though my ability to understand even extends as far as it does with you, I don’t understand his extremely simple explanations - the poor guy’s understanding is so consistently backwards.
I don’t have “man up” to address any argument, whether it’s one I support or not - I’ll address it fairly and rationally without partiality either way. I’d be quite happy to analyse any new arguments in this same way, but unfortunately I can’t quite get to the bottom of what you’re building here. Sometimes I feel like I’ve grabbed hold of something, but it goes away when I look into it a bit further.
In the seeming absence of some of your workings, and apparent jumps in your “notes”, I think maybe it boils down to some non-standard terminology that you’re using, like “convergence theory” (I don’t understand the exact process of getting the progressively different results that you’re presenting) or terminology that you’re using in a non-standard way such as the exact distinction “carrying left to right” and “carrying right to left”. Maybe an understanding of them will help me piece together these seemingly bizarre equations of yours.