Not just 10K, 10K unique and verifiable signatures. Except in Wisconsin of course. :’)
And they also have the at least 400 per congressional district rule as well.
To be honest that sounds irrational and capricious. It means that, if in one congressional district every person hates Rick Perry, and, if in every other congressional district every republican in the state loves him, then he can’t be on the ballot in Virginia.
10,000 votes sounds arbitrary and whimsical as well, and it requires time and manpower that every candidate has in limited supply.
I’m betting this could be challenged legally if someone really wanted to do so.