I've seen plenty of posts regarding how Rome or Song China came close to industrializing. I am inclined to discount these claims, however there is a hint of truth. Most notable comes with the somewhat vague and unverified claim that there was a proto-steam engine within the Library of Alexandria. and when this Library was burnt to the ground, it set back human history by one thousand years.
I am also inclined to completely discount this claim.
Despite the somewhat mythic nature of the Library, for example we don't even know when the place actually burnt to the ground (many historians believe there were in fact many libraries that burnt down and were subsequently rebuilt) and we don't really know what exactly it contained, it is interesting to ask the question: How early can an industrial society arise?
I see various candidates:
Rome, with its hegemonic status and its relatively advanced engineering and architecture could possibly have led to industrialization eventually, though I assume this would require more influence among the freed men and Plebes rather than the massive slave class that rose IOTL.
Ancient China, with its hegemonic status and large population base, though OTL shows that despite virtually unending wealth, industrialization never occurred in China until recently.
Ancient Greece had all the competing states and a culture of philosophy and learning. This is perhaps the most far fetched of the group, but Greece could potentially lead to industrialization, though there is a lack of resources for this to occur in Greece or its various colonies.
Medieval or Renaissance Venice, with its vast riches and nearly industrial ship production, I feel Venice is a very good candidate for industrialization since we know quite a bit more about their actual level of advancement compared to the information we know in the ancient world. All Venice is missing are the resources for industrialization and the advancement required to build basic machinery, but the extensive merchant class could make up for the former at least.
17th or 18th century Europe. All the steps are already taken here, this would simply be quickening the pace of technology so the industrialization happens a century or two earlier.
So, with all that being said, what is the earliest point where industrialization could occur?