Alright, let’s start this off. The Economy of Cities, chapter one: right at the most controversial part of Jacobs’ thesis (and, it seems like, the most separable part): She claims that the first cities predated agriculture.
Her argument is a just-so story, together with arguing that the mainstream view is also a just-so story. Her view of the mainstream, paraphrased:
Adam Smith, in the Wealth of Nations, observed that (circa 1770) countries with the least industry also had the least productive agriculture, and even in industrial countries the most productive agriculture was nearest the cities. He nonetheless asserted that agriculture preceded cities, and that only food surpluses from successful agriculture enabled specialization and the formation of cities. (Jacobs says he makes this claim largely due to taking the Bible, specifically the creation of Adam and the Garden of Eden, as gospel, and points out that this was significantly before Darwin and before the first findings that established that the world was very old, and so this kind of unquestioned assumption was not unusual.) Marx questioned and re-evaluated many of Smith’s positions, but not this, and since then both economists and anthropologists have taken it as given and cited the other group’s findings as the reason we are sure of it.
Though archaeologists are frustratingly vague about how they are so certain*, they were aware of her ideas pretty quickly and did not consider them worthy of comment. This paper much later summarized (badly) the evidence against the cities-first theory; the upshot is that animal domestication and plant crossing began before villages in some cases, and the latest they can be placed is cooccurring with the development of the earliest large-scale settlements, like the example Çatalhöyük. The differences are multiple millenia, using carbon dating times which are +/- a decade or two and strata-based relative dates with error bars measured in centuries at most, so unless the field collectively has its head stuck in the sand it’s a strong case.
EDIT: A recent survey accidentally discovered what appears to be the remains of a 9000-year-old city. If true, this derails conventional wisdom about city formation enough to bring Jacobs back to viability.
The general outline of Jacobs’s just-so story is refuted, but not all the details, so I’ll sketch it for you. A trading post for a tribe that harvests obsidian from volcanic deposits and their neighbors develops, and many tribes come from distant places to trade for obsidian, bringing nonperishable food (live animals, seeds and nuts) and useful trade goods. The permanent population start being a good trading site for disparate things, not just the original obsidian. The excess live animals and seeds lead to good conditions for developing planned agriculture and domestication. Eventually they have excess food from their own production and spin out herding and farming to their environment and along the trade routes, becoming a properly specialized city.
As noted, the story by which this is the (or even an) origin of agriculture is unsupported. But it does show how Jacobs’s core idea of import replacement could function for early cities and for agricultural technologies, which lead into the important and not discarded piece of this theory: medieval Europe. Medieval Europe (circa 1000 CE) had lost all the tools and methods of Roman agriculture; even the institutionalized monastery farms still kept mostly trappings despite their size. Romans alternated two crops between their two fields; medieval farmers reused fields for several years until they grew unfruitful and were left to lie fallow for several years until they burned it and retilled the ashes. When the three-field system of grain farming was invented (a three year rotation, wheat -> oats -> empty), it was first used in the interior farms of walled towns and cities and spread along trade routes. Roughly 600 years later, the innovation of planting alfalfa as grazing fodder in the third year, which was a massive increase, spread the same way, starting in city gardens in France. This has continued to the modern day.
My general takeaway: as far as the claim that most technological innovations start in cities and that this applies robustly to agricultural technology, it seems pretty solid. These examples are well documented, and others, in modern America (corn-fed beef and the machine hoe) and older (experimentation with animal domestication in ancient Egypt) provide reason to believe this is robust. But it did not extend back to the origins of cities, so possibly more investigation, which Jacobs probably did not do, would be needed to determine what regimes it holds over.
*As far as I can tell, it is ‘radiocarbon dating handwave handwave these are serious archaeological techniques handwave therefore agriculture is older’