I'd vote for a Well-Organised Alexandrian Empire Survives PoD - whatever it takes to build a functioning basically federal Empire that has plans for rulers and heirs dying under unusual circumstances. Rather than the Empire fragmenting, it stays intact, but with the rise of a Roman Republic and Carthage in the West, the Chinese and Steppe in Central Asia, and the Indian Wars in the SE, the Empire needs a huge army to survive, even if they do win in many circumstances.
As a result, the increased demands for manpower, plus the increased needs for a strong resource base leads to the application of more and more mechanical-power rather than slaves. The sheer size of the Empire brings hugely different intellectuals to Babylon, making it easily the worlds great intellectual centre, merging Greek, Indian, Iranian and imported Chinese thought.
After that, its a repeated cycle of desperately maintaining stability and unity - alongside the cycle of population growth, food and resource extraction innovations - and eventually the exploration to see if there are lands in the West. (I imagine a great eccentric King at some point).