Nope. When (OK,
if) a new president is elected, he will assume that role.
When we are talking about the people that are at the top of the system, there is no judge or adjudicator.
The Democratic system assumes that the players respect the system.
You have never had a case in USA where one ruling president hasn't accepted the results of an election and hasn't conceded. All of them so far have conceded.
What process is in place in USA if the losing President refuses to ever concede?
Would the Republicans in the White house, or the Senate abandon Trump and insist he concede? What did they do when he was impeached for putting pressure on Ukraine to help him cheat in this election?
Can the police step in? Remember Bill Barr is doing everything he can to please Trump.
What about the military? Trump is the Commander in Chief.
For the military to step in, it would be considered a military Coup. It would be unconstitutional.
The handpicked Supreme Court can make a judgement, but Trump doesn't have to accept that judgement. Trump would just go down that route to stall for time. He would never accept a judgement that goes against him.
It would need to be the people to protest on the streets. But Trump has many loyal followers. Do you think Trump's base will abandon him? Or do you think they will accept Trump's claim that the election was rigged? They will defend Trump against protesters looking to throw him out.
The only way this ends orderly is if Trump gracefully concedes.
Trump doesn't act with grace and won't give up power unless forced to. Who is going to force him?
It might even be the case that the only way Trump avoids a prison cell is by remaining in power.
Anyway, I don't know. I'm just interested to see what happens.