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Regarding reactors, I'm actually NOT sure China is capable of mass producing those types of nuclear propulsion modules. We have very seldom seen nuclear propulsion vessels from China, even ones which would make sense such as the 003. Even the small French CdG uses nuclear propulsion, if the existing reactor is so good, why wouldn't China procure more of it? If not for CVs, at least for stuff like icebreakers or even larger surface ships.
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The fact that China has not done "X" yet is not proof they will not develop the capacity to do so in the near future.
China has more civilian / commercial nuclear reactors under construction, 22 last count, than any other country in the world. Furthermore the state council has approved construction of 10 new reactors in the year 2022 alone. The USA only has 2 reactors under construction. Western Europe only has 3 reactors under construction (1 in France, 2 in Britain). I won't mention how badly over budget and over the dead line these projects are. The numbers are clear, "The West" is turning away from nuclear power. The future lies in the East. Within the next 5 years I expect there to be 50 nuclear reactors under construction in China. This will be more than the rest of the world combined. China will totally dominate the industry.
There is a relationship between commercial nuclear Technology and military nuclear Technology. Having a robust commercial sector creates a large workforce of human capital and manufacturing capital that can be diverted to military needs. I'm not an engineer but If the Chinese nuclear industry can build 1,000 MW commercial reactors, I see no reason why they cannot build 100 MW naval reactors and stick it inside a submarine or aircraft carrier. I'll take it one step further. It was the Americans that invented nuclear marine propulsion. I believe the Chinese will ultimately surpass them.