The assumption is based on a pretty good amount of historical evidence.
Apart from the one Qing China example, the rest of the supposed evidence utterly escapes me.
(which also explains why Europe's geography makes it difficult for continental empires like the one you're proposing to thrive).
First, the argument is made questionable by counterexample of the Roman Empire. It lasted several centuries without experiencing so terrible geographical difficulties, and a compelling argument could be done that one main reason fro its downfall was that it did not expanded enough, to its natural boundaries when it failed to conquer Germania.
Second, this is not really a valid argument for an Empire like the Carolingian one, which is centered on the Franco-German-Italian bloc. No real geographical obstacle exists to stop an empire from thriving in the great northern European plain from France to Poland-Hungary, once the area is properly settled to an agricultural-urban economy: plains, good land, temperate climate, navigable rivers that provide transport for trade much more than they are obstacle. The bloc only has a real geographical obstacle in the Alps, but sincerely, if the state is strong, its is far from an impossible one: European history shows that either France or Germany has always ben able to control Italy effectively, if the power is cohese, barring superior opposition from another great power.
I freely concede that a continental great power would have serious problems expanding its control to the periphery (British Isles, Russia, Balkans) as Napoleon and Hitler discovered. And indeed I have assumed that our united CE would fail (or avoid trying) to do so. When the CE accomplishes the Reconquista, it would add a second significant geographical barrier. Would it be able to integrate Iberia too, and complete the Franco-German-Italian-Iberian core, or would Iberia escape its grasp ? Hard to say. Roman precedent says it can be doen, Napolonic precedent says it cannot. I can see political and military butterflies either way. But sincerely I do not think there is nowhere good groudn to argue that geography makes the political unity of the carolingian core untenable.
I'm paraphrasing here but basically he says that the more different empires you have the more probable it is that one of them will embrace new technology or ideas.
The argument may justify the assumption that poliitcally-fragementated areas may (but it is not a given) progress faster, not that big centralized empires are doomed to enter long-lasting cultural stagnation or shall necessary progress slower. Moreover, it fails to take the economic and cultural damage that political fragmentation causes into account. It's hard to see how barbarians sacking cities manages to accelerate the progress of mathematics.
Diamond uses the example of Christopher Columbus to illustrate this. Before Christopher Columbus led his expeditions to the Americas he visited and was rejected by the King of Portugal twice, was refused an audience with the rulers of Genoa and Venice and was initially turned down by the Spanish before they finally agreed to fund his voyage.
Diamond is utterly forgetful of the fact that there were very good reasons why Columbus was repeatedly rejected. His arguments were *wrong*: he had completely bungled his calculations about the cricumference of Earth and the distance between the Atlantic coast of Europe and the Eastern coast of China, which was much greater than what he thought he was, and the travel completely undoable with 15th century naval technology without a stop in a continent he did not knew existed. The kings that refused him had his calculations checked, the error identified, and he was earmarked like a crackpot. His discovery of America was completely unforeseen by him, and he was unable for the rest of his life to recognize he had accidentally found a new continent. He was basically the equivalent of a pseudoscience guy that pesters the White House or the European Commission trying to get funds for a perpetual-motion machine. There was nothing obscurantist in the refusals that Columbuds got. The example is irrelevant to prove the argument.
In this TL yes it is possible that the Carolingians fund some kind of Columbus figure, but it is just as possible that they choose not to and he has no place to go, or, even worse, they take the advice of some other idiot who says the world is flat and have all ships large enough to make any oceanic voyage destroyed.
First, in this TL a Columbus figure (let's assume that he is someone with a scientifically valid evidence and not a stubborn lucky crackpot like OTL Columbus, so there is no obvious reason why he's refused; say he has evidence about the Vinland route) may still go to the king of Britain-Scandinavia if the Carolingian Emperor refuses him, or viceversa. Second, if he gets a refusal, he (or his younger relatives or business associates) could have better luck with the next Emperor. The probability that the Empire is going to have an ongoing string of obscurantist rulers grows increasingly smaller. Third, there is also plausibility limits to such obscurantism: to use your example, no one that argued the world was flat would have been taken seriously by any educated Middle Age elite person.
I'm not saying you have to have the bigger = stagnant assumption hold true in every TL but it is something to think about.
What I can think about, given a review of historical evidence, is that on average, big centralized empires and politically-fragmented areas most likely may progress at roughly the same speed, sometime fester, sometime slower, given various butterflies, and "smaller=better" is a faulty generalization clichè that is built on little more than the Qing China special case.
Yes, a continent spanning empire could make all the right decisions and end up with a man on the moon by 1600 (and this kind of wank does have an audience), but most of the more experienced posters expect there to be more balance with ups and downs.
But this is quite different, and much more reasonable. The issue here, is that continent spanning empire, all other factors being equal for cultural progress, deserves the good faith expectation of being able to put a man on the moon in 1925, or in 1998, and not be doomed to use horses and matchlocks in 2009 just because of its size.