ASML owns the company that builds the light source. They acquired it, it's a US company, which is why US export controls apply, that's all. If needed, they could replicate the subsidiary in the EU.

This is too far from correct for any correction to be anything but a full restatement of the facts. Moving the tech over requires US approval. Listen, the Dutch are not going to risk it. Even if they were, ASML would not risk it because all of their customers wouldn't buy anything from a company that's on the EAR Entity List (which is where they'd end up if they tried this without the US allowing it) without US approval. I don't get why people are saying this stuff. It's like saying "Oh yeah, so you divide by zero and then multiply both sides and ta-da". Like, the whole statement is nonsensical.

To enable the whole thing to work you'd need the US to have shrunk to the equivalent of Canada in influence. I'm not saying that's impossible, but in that scenario, the Dutch might well be trying to keep Russians out of Amsterdam and the Turks out of Germany rather than trying to pull an IP heist on the Americans.

You can buy an e-book on Kindle and Amazon still controls what you do with it, right? ASML's ownership of Cymer is like that, except it's the US instead of Amazon.