i've been over this before, but briefly.

while there have been some attempts to link kurdish ethnogenesis further back in time, both the origin of the term (which means 'tent-dweller' or 'wanderer'.....which i'm interpreting as refugee) and the proposed date of linguistic differentiation from persian suggest that kurdish ethnogenesis did not actually take place until well after the islamization of the fertile crescent. stated very tersely, when the dust settled after the major shift in power that happened after the arabs moved north and (1) kicked the romans out and (2) flat out conquered the persians, there were all of a sudden all of these iraqi persians living in the mountains, and there's really not a story as to how that happened - kind of like how there's really not a story about how we ended up with a really large amount of jews in eastern europe.

we know that there was nobody called kurds in the region during the classical period, and we know that persians had been in control of iraq almost continually at that point for roughly 1500 years, with the only substantive break occurring in alexander's lifetime; the post-alexandrian seleucid state was eventually overrun by parthians from central asia, but it was essentially a continuation of the achaemenid state. so, the persians had a very deep history in iraq at the time of what was initially an arabic military occupation. so, we know that there was no such thing as a 'kurd' in any of the literature from the time of cyrus (c. 550 bce) until roughly the year 900 ce, and we know that the persians dominated the region for almost all of that period, until about 650 ce (roughly, and the dates are actually a little blurry).

then, we have these kurds living in the mountains like refugees, speaking a new dialect of persian that just pops into history just right then. further, we have a lot of muslims in mesopotamia, all of a sudden, living in baghdad rather than ctesiphon or seleucia (initially babylon). over time, we see these kurds develop as a kind of ruling elite, as well. if you look into it, you realize that the entire babylonian renaissance that was centered on baghdad was really kurdish, rather than arabic, in origin.

i don't even know how you'd convincingly prove the hypothesis, but it just seems obvious to me to put two and two together and deduce that quite a lot of iraqi persians must have ended up in the mountains around the time, as a way to evade what was happening in the city, and surrounding areas. and, all you have to do is look at what happened in the isis invasion to get a kind of rough idea of how that must have happened - the arab groups come in from the south in a rage, and the persians must have sought higher ground to get out. a few hundred years later, they have a new dialect and a new identity. or, sort of; if you can interpret being called a refugee as an identity.

what that means is that, if i'm right, the kurds are essentially a 1300 year old iraqi persian refugee population.

yes...

1300 years of being refugees in the mountains.