They could just ignore Trump as he has no authority to so limit a private company.

The labs will not just ignore the order, there are too many other levers they can try to pull to mess with those companies. Just for some examples, think about the number of employees reliant on visas that could be revoked, the government contracts that the hyperscalers hosting them that could be canceled, the certifications that all the data centers need to be hooked up to the grid, the tariffs that could be put on critical components, the IPOs that need to be approved by regulators, the bill introduced in Congress to seize 50% of their equity...

Lots of these moves would and should be struck down in court as an arbitrary and capricious use of administrative power. Some of them might not be, and in the meantime you're signing up for tons of trouble. A trillion-dollar company does not simply go to war with the US government.

A more mid-sized company that's not so intertwined, but not so small that they can't get a good legal team, might be another story.

Ignoring US export control laws that have been on the books for nearly 50 years is a good way to pay millions of $ in fines and/or land in pound-me-in-the-ass prison.

Ask every satellite launch company in the 90s how that worked out.

These regulations have been in place since the 1990s and have been applied by every administration. It isn’t a new authority and many companies have had the opportunity to fight it. Anthropic’s lawyers will know this.

Yes he does. They could ignore the US government, but will likely quickly find themselves in court fighting a fight that they are likely to lose and isn’t worth fighting anyway.

Aside from how brazen and stupid this would be, the executive branch does have ways of limiting them and their sharing of tech, as we've seen.