Allow me to shift focus somewhat, but not much.
You mention Pure Logic. Pure logic as a concept has an origin.
I did some tracing and was surprised to find this via: https://books.google.com/books?id=dWp2HUQl-0UC&lpg=PA149…
While our current Western histories regard Aristotle as the father of Logic, it was Alexander of Aphrodisia who first mentioned it, 500 years later. Aristotle considered it the Analytiks and not the concept Logic as we know it now.
I was initially going to mention that in your thought experiment as well as your explanation, you spanned the entire range of history between origin and present for where there was divergence between the disciplines, but what I am seeking is their points of convergence. [both geometrical terms I believe, metaphorically used, as is the term point, but using geometrical terms tends to lead credence to concepts when expressed in English].
But then, I discovered ^^ which was surprising to me.