The Sao Francisco is a long river that flows from the southeast to northeast region of Brazil, linking the southeastern mountains to the northeast coastal plain via the dry cerrado and even dryer caatinga environments. Historically, it was navigable for much of its length, making an important contribution to colonial Brazil's economic development. While much of the Sao Francisco river basin today is sparsely populated, there are some important towns along its length, some of whom rely on irrigation from the river for agriculture.

My question is, would it be possible for a pre-contact Native society to develop an irrigation system to enhance agriculture using this river? Bonus points if the resulting agricultural system is productive enough to support a highly stratified society, something equivalent to the Mali Empire or the early empires of the Middle East by the time of European contact. Or is the river too unreliable, or the soils it goes through too unproductive? If not, what barriers might have prevented the rise of such a polity among the Native Brazilians of OTL?
 
I have absolutely no data whatsoever on native Brazilian population density prior to the arrival of the Portuguese, so I unfortunately can't tell you anything other than that I find this idea of yours extremely interesting.

Personally, I think the best place in Brazil for a "hydraulic empire" of the kind you're describing (besides the Amazon basin, which had places like Kuhikugu) would be in the state of São Paulo. The climate there is far more forgiving than the northeastern caatinga, and the Tietê River should provide not only arable land, but easy transportation as well.
 
Maybe with a proper irrigation system the São Francisco river could be like the Nile of Brazil. Unfortunately this region is very isolated, perhaps an alternate Tupi migration following the river could bring new crops and contact with the broader Tupi world
 
I wish I knew more about the archaeology and societies of this region, since this idea does have potential. It's curious there wasn't a larger civilisation there, since the area seems to meet the definition of the "circumscription theory" of a narrow band of good farmland (albeit one which needs improvement) surrounded by less valuable land like mountains, swamps, and jungle. And like in many cases of state formation, the diversity of nearby land is also helpful for developing a state since it gives maximum diversity of resources and trade goods. This case would almost be akin to the Columbia Plateau, which was also a good place for a hydraulic empire/early state society based on circumscription theory yet also where there were no state societies.

I wouldn't be surprised if there was some irrigation, but it might have been an issue like in Oasisamerica where the society never fully developed and was smacked down with droughts and issues with soil salinity that resulted in more complex agriculture and monument building being largely abandoned. Or perhaps there was never a push, like how the Patayan culture of the lower Colorado River (and their Yuman-speaking descendants in later centuries) only farmed for part of their food (despite the vast theoretical productivity of their land) because the floods and droughts were not reliable and it was deemed better to not devote too many resources to their fields in favour of hunting and gathering.

Looking into it, there is also some indication that the caatinga was a managed land based on artificial fires for slash and burn agriculture (and probably hunting and gathering too). Such land use implies the area in the more distant past might not have been as suitable for a hydraulic despotism type society to arise. Most areas of the caatinga appear to receive at least 300 mm of rain each year on average and usually more than 400 mm, which is more than enough to support dryland farming (precarious as it may be due to drought), therefore slash and burn agriculture was certainly viable.

Overall I'd say it would be an interesting addition to the civilisations of the New World, directly east and south of the enormous cities of Amazonia and on the opposite side of the continent from the powerful kingdoms of the Andes. How it would look, I don't know. Maybe like the Purepecha (who live in a similar semi-arid land) it would be a sort of palace economy with very centralised rule, but with the sheer advantage that river transit brings on a continent with few domesticated animals. They probably would mostly grow cassava, maize, and sweet potato as their staple crops.
 
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