I believe what I said (in #1179 to which you were responding) is "The paradox is then "solved" by assuming that the first cause is uncaused, and "exists" just because, for no cause or reason whatsoever, as a logical necessity."
If there is any reason lacking then it is in asserting that the uncaused "exists" if cause is the reason for existence. Therefore the uncaused (first cause) can not logically exist (there is no cause for its existence) even though it "must" [sic] logically exist to stop infinite regress.
Aquinas was thinking inside the box, and so if everyone else who does not allow for existence to be eternal, repetitive, self-causing, etc. This is a lot more rational than implying some "thing" that exists in eternity, without change, or cause, and then decided (!), without a cause of course, to "step outside" the eternity and start creating without itself undergoing change!
A logical necessity is otherwise known as a reason.
The first cause argument is a logical argument - the very opposite of "for no reason."
If you have a similar syllogism that works, you know I'm going to ask to see your work.
:)