No, TSMC is equally scarce in the US and PRC, and also equally absent from Brazil, Nigeria, and Malawi, so it's not just in reference to TSMC, though they are indeed one indicator.
The Romans seemed to innovate because they were conquering the Greeks and adopting their innovations. When they had no more Greeks to conquer, the innovations ended.
The Chinese did innovate by applying gunpowder to guns; that's why they've had guns since the 12th century, while the oldest European guns are from the 14th century, evidently copied from the Islamic world, who probably copied them from the Mongols, who probably copied them from the Chinese.
Your history is missing about seven centuries between the Dark Ages and the Enlightenment; you might want to read up on https://en.wikipedia.org/wiki/High_Middle_Ages, https://en.wikipedia.org/wiki/Late_Middle_Ages, and https://en.wikipedia.org/wiki/Early_modern_period. Also, the Roman Empire didn't finish collapsing until 500 years after the end of the Dark Ages; you may be thinking of the Western Roman Empire.
I think there's a pretty continuous stream of major innovations since the beginning of the Iron Age, though it's true that often one or another culture has become a backwater and stopped innovating, like Europe during the Dark Ages, or arguably Yuan China. I think the difference between currently existing cultures is more phase-change-like, in the sense that intellectually ambitious Brazilians and Nigerians tend to move to the US or Europe.
To take the particular example of Descartes' geometry, following Euclid you had Archimedes, Apollonios of Perga, Pappos, Zhang Heng, Liu Hui, Aryatabha, Bhaskara, Zu Chongzhi, Brahmagupta, al-Mahani, ibn Qurra, al-Khwarizmi, Shen Kuo, al-Haytham, Khayyam, Yang Hui, Pacioli, al-Tusi, Tartaglia, and Viète, among numerous others; Descartes could draw on all of their work, much of which had been translated into Latin, and other parts of which had influenced later writers whose work had been translated into Latin. Bombelli invented complex numbers a century before Descartes!
https://en.wikipedia.org/wiki/Timeline_of_geometry and https://en.wikipedia.org/wiki/History_of_geometry go into more detail. No century passed without major advances in geometry, though many centuries pass in which none of those advances bear European names.
And we could make similar lists about chemistry or algebra.
The explanation to me seems straightforward: the Roman Empire conquered the civilized parts of Europe, the Middle East, and North Africa, and it was evidently not a hostile environment for innovation in general and geometry in particular. Consequently the only Roman name in my long list above is Pappos. The Roman yoke was then supplanted by the heavier yoke of the early Christian Church, which actively sought out and destroyed knowledge; to them we owe not only the Dark Ages but also the loss of the knowledge of Egyptian hieroglyphics, the burning of the Nestorian and Arian texts, the desecration of the Archimedes Palimpsest, centuries of persecution of the Jews, and the suppression of heliocentrism, as well as the burning of the Mayan codices and the khipu, losses whose magnitude will remain forever impossible to calculate.
Though the book-burning of the Christians evidently goes back to Paul, such anti-intellectual movements are not unique to Christianity: Qin Shi Huang, the Nazis, Pol Pot, the Boxers, Comstock, and Boko Haram provide counterexamples. The Christian atrocities are greater in magnitude only because the Christian Church ruled so much of the world for such a long time.
So why did the Dark Ages end so soon, given that the Roman Catholic Church still plays a strong and sometimes dominant role today, 1000 years later? Perhaps the creation of the Roman Catholic Church in the East-West Schism is one important cause (breaking in half the Church that had remained undivided since its genocide against the Nestorians and Arians), but it's an enormously debated question. Maybe Petrarch, the guy who first called the Dark Aages "dark", played a big role in ending them. Or maybe the tyrannical nature of Christian rule had weakened its victims until they could no longer suppress learning and innovation — nor resist conquest by more highly developed cultures, as in al-Andalus, the Seljuk Empire, and eventually even Constantinople itself. Or maybe the Crusades that followed increased contact between the Christian world, divided between decadent Byzantium and fanatical Europe, and the literate world. Or maybe it was the Medieval Warm Period. Or the Plague.
It's difficult to say in part because we know so little about the Dark Ages. What horrifying abomination enslaved the minds of Europe so terribly, despite the Greek legacy of great intellectual achievement? What were the contours of day-to-day life? How did people think about law, war, dispute resolution, rulers, legitimacy, slavery, heresy, workmanship, innovation, religion, and learning? We know that both cenobite monasteries and Jewish communities survived in Europe throughout this time as enclaves of learning, but the texts they wrote during the Dark Ages are mostly copies of earlier texts; they do not give us much insight into these questions. So what was the cause of this stagnation and backsliding? Historians disagree.
A different way to look at the question is that perhaps the stagnation and backsliding of the Dark Ages is not the part that calls for an explanation; perhaps progress and innovation is the exception and collapse, or at least bare subsistence, is the rule. In this conservative view, societies that can sustain geometers, mechanical inventors, and chemists are fragile and delicate things; such intellectual progress requires a rare social climate (though proponents disagree on the exact preconditions) and can easily be snuffed out. And certainly in history it is easy to find not only examples of such snuffing but also peoples who have remained relatively undeveloped for many centuries without ever developing.