I don't see any formal definition- certainly not in the first 7 sentences. I don't actually see any definition at all. I see a lot of general vagueness and use of undefined symbols.
You say "A single-simultaneous-connection is any single real number included in p, q
( = D = Discreteness = a localized element = {.} )."
What do you mean by "included in p,q"? In particular, what do you mean by "p, q"? I would tend to assume you mean "any of [p,q], [p,q), (p,q], (p,q) which you had given above. I take it then that "A single-simultaneous connection" is a singleton set?
"Double-simultaneous-connection is a connection between any two different real numbers
included in p, q , where any connection has exactly 1 D as a common element with some
other connection ( = C = Continuum = a non-localized element = {.___.} )."
Okay, so a "double-simultaneous-connection" is a pair of numbers?
"where any connection has exactly 1 D as a common element with some other connection" is not clear. You appear to be saying that two "connections" (I take you mean "double-simultaneous-connection") that have both elements the same are not considered to be different. That's actually part of the definition of set.
I have absolutely no idea what " = C = Continuum = a non-localized element = {.___.} )" could possibly mean.
"Therefore, x is . XOR .___."
This makes no sense. The only use of "x" before this was as a bound variable in the (standard) definition of [a,b], [a,b), etc.
In any case, you have been told repeatedly that your use of "XOR" has no relation to the standard use. Please don't use a standard notation for a non-standard use.
You seem to be still agonizing over the difference between the discrete integers and the continuous real numbers. I can only suggest again that you take a good course in basic mathematical analysis. (And it might be a good idea to learn what a "definition" really is.)
By the way, what does "non-commotative" mean? Did you mean "non-commutative"? I didn't see any reference to that in you post.