OP- hugs.
You need a sit down with your kid; unlike some of the other posters, I don’t think I’d be asking, AGAIN, did you cheat. Asked and answered- he either did or didn’t, and continuing to ask the question means you guys aren’t moving on-- and you’re either proving to your son that you don’t believe him (if he didn’t cheat) or asking him to lie (if he did). And I don’t think either of these options are where you want to be in your relationship with your son.
What would I do? Make yourself a list of desirable outcomes from this situation, and start there.
Outcome number 1- a good, honest relationship with your kid.
Outcome number 2- restore your son’s self-confidence
Outcome number 3- Not have this incident ruin the rest of his year, or poison his relationships with the faculty and administration.
Outcome number 4- Establish good boundaries with your son AND for your son.
So my game plan (of course, your outcomes will be different from mine, so your game plan will be different…) would be to assure my kid that I’m no longer interested in what did or didn’t happen. My only interest is in assuring him that whatever mistakes were made (and surely, copping to something he didn’t do was a mistake, but an understandable one), it’s not a big deal in the grand scheme of life, he didn’t push an elderly person down a flight of stairs, or sell crack to a third grader. It’s not life altering in any way, and you’ve already moved past it and so should he. Stuff happens.
Encourage him to move his seat in the class, and to make an effort to show his work on tests, quizzes, and homework. It’s a good habit to get into even if he intuitively understands the problem, because when math gets much harder (and it will, if he continues to like math) it will help him isolate errors if he gets into the habit of outlining his work. It’s also a great discipline for other subjects once he gets to college- philosophy, any foreign language if he’s translating dense and complicated works, engineering. He may have group projects in college and so getting the answer is no longer the goal- collaborating is. He could be on a team trying to create new and cool stuff- and showing his work is the way the team will be able to feed off each other’s ideas.
Encourage him to approach this as a speed bump- and move on.
Once you see he’s moving past it- sitdown number 2. Explain that if he is EVER in trouble, he does not need to go it alone. If he’s arrested- he needs to ask for a lawyer immediately and then shut up. It’s his constitutional right not to answer questions without a lawyer. If he gets into academic trouble, he does NOT need to answer questions from authority figures. There are parents (while he’s a minor) there are ombudsmen at college, and of course- more lawyers. Explain that it’s natural to be intimidated by aggressive questioning, and he should understand that calling in reinforcements is NOT admitting guilt- it’s having an unbiased pair of ears sitting with him who will not become emotional and can help him sort things out.
You’ll get past this. Take the grade hit without a protest so your son can move on.