Well the point about integration is that, as you know, Western countries, and esp. new world Western countries have accepted succeeding waves of immigration. The integration into general society of these immigrants is, presumably, important, because if individuals are not properly integrated society's health will suffer as a result. Now, if it was the case that an immigrant's race was a strong predictor of an individual's future integration, then trying to convince people that individuals who integrated successfully in part thanks to their specific race (say, race A) are in fact not of that race at all in order to convince people to accept immigrants from other races, would be a deceitful tactic, since it would make people think that being of race A does not contribute to an immigrant's ability to integrate, when in reality it would be a significant factor, and so this might convince people to accept large amount of immigrants of different races (say, race B, C,) who might not be able to integrate well at all in part because they are not of race A.

Now, regarding the definition of ''white'', I think you're making this more complex than it should be, and I suspect this might be because you've internalized leftist historiography of right wing racialist ideas (like when you mentionned that Irish are not white). Nobody here, I'm pretty sure, agrees with the ''one drop rule'' or think that if you're 1/18 white that suddenly makes you white. Clearly, if a thing is made up of two or more than two parts, for you to consider it mostly made of one part, that part would have to represent 50%+ of the thing, right? And ideally, something in the 70-80% range, correct? If you ordered a custom-made table, one half made out of birch wood and the other made out of oak wood, you would probably simply describe your table as half-oak and half-birch, but if it was 60% birch and 40% oak, you might be tempted to call it a birch table with a good portion of oak wood in it, and if it was 85% birch and 15% oak, you'd probably call it a birch table with some oak wood in it. So to be considered simply ''white'', you'd minimally have to be 50% white, and probably everyone would agree you were white if you were 85% white. There is some interpretative wiggle room here, but it's not much, clearly nobody can think of you as mostly white if you're 10% white, and basically nobody would consider you non-white if you were 90% white. So is Manuel Blum white? I don't know, we'd have to look at his ancestry, but looking at him, he seems to be mostly white, and so he probably is. Unless you have access to his genealogical record, or you think his physiognomy doesn't seem European (physiognomy is an imperfect proxy for racial classfication, but it works well enough), then you'd have to agree with me here.

''You haven't actually shown such a disparity''

In terms of scientific discoveries and inventions? Well, no, because to conclusively show that would be an herculean task, but I don't have to be held to such a standard in the context of this discussion. It's common knowledge that some places were hotbed of civilization and some places not so, that Europe shot past everyone else starting from 1500 (and according to Comin, Easterly, and Gong (2010), Europe was actually not lagging behind the Middle-East during the Middle-Ages) and that this resulted in a majority of today's inventions and scientific discoveries resulting from the work of white Europeans, simply because many more discoveries and inventions were made during 1500 to today than were made before that point, even if you added up everything up, at a time when Europe became world dominant.

Now I'm not too sure what this reply chain is about ; if it's about European contributions to science and technology, then it's basically uncontroversial to say that they're very important generally and especially so starting from the 1500s, where Europe became dominant, and I think you'd basically agree with me here.

If it's about what the alt-right thinks of race, you can read stuff from here http://www.humanbiologicaldiversity.com or here https://www.unz.com/jman/hbd-fundamentals/.