You presume incorrectly. Any coordination would make Elon and his allies a “group” which would mean they are collectively treated as an individual.

How do you prove people are in a group of this nature?

You check to see if they are communicating.

How do you that without a search warrant?

Ask, I guess. The people doing the communicating are perhaps committing fraud and endangering their own wealth and freedom if they lie. I'm willing to bet the SEC has regulations that apply to this situation already.

If three large shareholders came in with a similar agenda but did not disclose group status, the SEC would investigate. First, the SEC would request that they turn over communications records. 99.9% of the time, they would. If they didn’t, the SEC would seek a warrant.

Again, first lines of the shareholder rights agreement. This is something the lawyers thought of approximately 3 minutes after poison pills were invented.

https://www.law.cornell.edu/cfr/text/17/240.13d-5 seems to indicate they have to be explicitly coordinating to be legally defined as a "group" for this purpose (and Twitter's filing highlights this definition), which seems hard to prove. Twitter can't issue a search warrant, the participants would presumably not write any such deal down on paper, and the action is readily explicable by "I think the acquisition is gonna go through and he's offering a premium, so I'd profit".

The financing is in 3 pools. If each could buy 14% and Musk increases his stake to 14% without triggering the poison pill, that’s 56% in total. If they had 56%, could they immediately gain control? This is getting interesting.