They have not; a social media post does not satisfy the requirements of 10 USC section 3252.

They are required to notify Congress (they have not), prepare a report with specific sections (they have not), and the reasons must fall within a set of categories outlined by statute (this does not).

There will be a court fight and they will lose, just like they lost the tariff battle, because of poor competence.

(Trump's post on Truth Social was actually fine. He said the USG would stop doing business with Anthropic, which is within its legal right. Hegseth's follow-on post is the problem. It is possible that Trump did not expect or want Hegseth to do that, that this was meant as bluster to bump along the negotiations; Hegseth has a recent history of stepping out of line within the administration and irritating people like Rubio.)

That's part of the recurrent confusion with this administration. In previous administrations, including Trump 1, people didn't need to spend a ton of time thinking about what it means to make a legally effective proclamation, because there was a baseline of competence. When a government official announced "We're doing X", they would do so as a summary of a large amount of legal process with the intent and effect of causing X to be true. If you went to challenge it in court of course, you'd have to identify some specific action as the label, but everyone would understand that this is a formalism.

Here, Hegseth has simply made a social media post. He did not publish any official investigation which led to the report. He did not explain what legal power would permit him to impose all the restrictions the post claims to impose. There is not, five hours later, any order on an official government website about it. So we have a real question. If a Cabinet secretary posts "I am directing the Department of War to designate...", does that in and of itself perform the designation, or is it simply an informal notice that the Department of Fascist Neologisms will perform the designation soon?