Apparently my hobbit metaphor from 2014 is now all over the place because Moldbug signal-boosted it, and damn if that isn't funny.
On a practical level, note that if I'd continued to use it regularly myself, this event would have been blocked. If you want to do this sort of thing on purpose, neglect is part of the strategy. Anyway, let's talk about the blood.
If you know how many aristocrats there are, how many children they have, and the proportion of aristocrat children who end up being hobbits, you can work out what percentage of hobbit-born children would result in more than half of all aristocrats being hobbit-born.
Let's say, for the sake of argument, that aristocracy is the top 1%. To make the math easy, let's pretend everyone has the same family size, and all 100%, every aristocrat baby, is themselves an aristocrat. A bunch of things cancel, meaning the threshold is a simple 1/99 => if more than 1.01% of hobbit children are aristocrats, at any given time, more nobles are lowborn than highborn.
Note that this is all linear or close enough, so modifying the number is trivial. If only 80% of aristocrat births are also aristocrat, then multiply the final number by 0.8. If aristocrat generations are half the size of hobbit generations, then divide the number by 2. Etc.
Note that the 1%, if we're talking IQ, is 135. I consider that midwit. You need ~130 to have what I consider basic literacy; these folk might see a reference and actually read it. If you tell them to google a book, the book might actually get read. "This is the only group that can identify from search results a book
suggesting that the claims made both for and against genetically
modified foods are unreliable." However, that's their limit. These folk don't do nested references - if a citation itself has a citation, it's not going to be read. They can produce original thoughts only through tremendous luck; their ideology is scavenged, not constructed. If it's not in a book already, they can't think it. The innovation-class IQ is distinctly higher than 135.
Although aristocracy is not merely IQ, from IQ you can calculate the actual percentages and how they vary by innate social status. (I estimate the real percentage is something like 30%, not 50+.) Regression to the mean is more aggressive the further from the mean you are. If you're curious, get your lab rats to crunch the numbers. Someone will have to find out how many children aristocrats have, though. Nobody has gathered that information yet, because they won't admit that noble/peasant is a real distinction.
Note that while it is very plausible that most nobles are lowborn, you don't get lowborn dukes and princes. You get lowborn squires and barons. The Gaussian has exponential decay.
Even if a duke or prince were born to a hobbit family, their distinctly mismatched environment is apt to do tremendous damage to them; you will get a ruined bastard prince, not a healthy prince. Especially in a tremendously envy-positive Fascist country.
P.S. Since the odds of a duke giving birth to another duke - even if married to a duchess - significantly differs from 100%, primogeniture is a terrible idea. Basically Moldbug is right about stockholding.
I would do something like distribute stocks of the estate evenly between the children, and they can work out succession amongst themselves via the buyout dynamic. Those who want it more can bid higher. If the non-inheriting children get something out of the deal (aside from their innate blue blood), it dramatically reduces the pressure they feel to commit fratricide. It no longer codes as, "Daddy loves one son 100% and all other children 0%." Bonus: they can buy out the father and have him step down before he goes senile and ruins the place. Anyone who deserves to rule can get the money without having to first inherit. Dukes especially should do it for free, but why not provide a healthy encouragement?
It also helps if you have them read Son Also Rises. If you strip a billionaire of every cent they own, after a decade they will be a billionaire again. If you grant a millionaire a billion-dollar business, it won't be a billion-dollar business for long. The world is fundamentally a just place; men of quality are rewarded, and men of lesser quality receive lesser rewards, even if someone tries to thumb the scale. Whether they inherit or not largely doesn't matter to their final fortunes, especially if the country isn't in the Malthusian trap. Remind them that mortal law can be bent. If their ambition is illegal, that's nothing but an inconvenience.
Goes double if you can buy out the estates of other people's children.
P.P.S. Given that innovation-class IQ is distinctly higher than 135, estimate how high Greece's average IQ must have been to produce Socrates, Plato, and Aristotle close enough in time to have all met in a single pub, using a an effective source population of well under one million.
P.P.P.S. By far my favourite race in Master of Magic has always been the dark elf. If you're too boring to appreciate this, my condolences.