Just adding that: That cultural matrix was a point of departure for Rome, and Rome (including the latter Roman Church) was an adaption of the above cultural matrix which then developed independently, distinctly, and now we know it as Western civilization. Western intellectual circles (from beginning) declared themselves from and for the Greeks, but more honestly considered, West is essentially a Roman affair. In any event, until the advent of Islam, there was continual and pervasive mingling of these core cultures and use of the shared language of Aramaic, going back to at least a thousand years before (circa ~500BC).

"West is essentially a Roman affair"? I don't think that's true at all; intellectually, Roman civilization was almost entirely sterile, appropriating the Hellenistic cultural heritage and then entirely failing to progress for centuries, even backsliding in many ways. And, despite the importance of Aramaic, Greek was a much more important lingua franca, continuing to be so until the fall of Byzantium.

It's true that the division between the Eastern and Western Roman Empire created a new East/West divide approximately corresponding geographically to the old Greek/Persian divide.

If you read history, the original East/West split was Persia/Greece.

Strange then to find (~Greek) Gnostic writings that have the Pythagores and Zaratustra attending Babylonian seminars, with latter teaching the former 'astronomy'. Archaic, no doubt, but put yourself in the place of the reader (in the "West") at the time of the writing and ask youself: What "divide"? Political order differences? East Asia was and remains far more 'alien' to Iranians that Greeks. Note that the East Asians who do not have a dog in this fight, typically group us all together and don't make the distinction.

I don't think it's strange at all; dividing the world's people into two geographical groups doesn't imply the groups have no contact or interchange! It's well established historically, for example, that Egypt and Mesopotamia developed geometry long before the Greeks, and Pythagoras is said (by nominally historical accounts, intended to be more literaly factual than the Gnostic parables you're speaking of) to have studied in both places. Both Egypt and Mesopotamia were also part of the "East" for the Greeks until Hellenistic times, even though Cairo is geographically further west than Cyprus; maybe Cyprus was also "East"?

Also, to the Chinese, India was "the West". The famous novel Journey to the West is a fictionalized account of the Buddhist monk Xuanzang who traveled from China to India to bring back copies of Buddhist texts that had not yet reached China. And nobody doubts that Buddhism originally reached China from India ("the West").