Regan's algorithm cannot prove that you did not cheat; it can only fail to prove that you cheated.

The question that Regan's algorithm is this: "Did you cheat?"

If the algorithm comes back with "Yes," then you cheated. But it cannot come back with "No." It can only come back with "I failed to prove that you were cheating."

You need another hypothesis: "Did you not cheat?" But this is an impossible test.

Agreed, this is also likely a high precision, low recall approach to avoid false positives ruining a player's career

However, allowing cheaters to get away with it could eventually ruin chess — that is, ruin all players’ careers.

A delicious conundrum.

Like a bloom filter!

I believe Fabi, but the question must be asked: how does he know (with certainty!) that the player was cheating. Did the player confess? Was he blatantly looking on his phone in front of him?

Imagine I see someone after an event, flushing evidence of cheating.

I am completely certain, because I saw the evidence myself.

But to anyone else? There's no hard evidence - just he-said-she-said. Maybe I was mistaken - or trying to get my rival into trouble.

Knowing how it would look, maybe I keep my mouth shut rather than throwing around wild, baseless allegations about cheating.

Maybe it was Fabi!

This seems like the most obvious answer but social dynamics make it hard to suggest.

It's suggested here now :) More spicy would be if Kramnik was cheating in his match against Topalov.

Making moves perceived to be outside his skill class I assume.

If it's that, then it's pretty arrogant - I mean, that's like saying: "he's too stupid to make such a good move, so he must be cheating"? Even (relatively) bad players are bound to make good moves from time to time, either through dumb luck or through occasional sparks of inspiration, so that's hardly a reason to be 100% sure they are cheating...

There are moves and lines that only Super GM’s will calculate. Not IMs, not FMs, not GMs. It’s not arrogant, it’s the level of play.

When you are #2 in the world at something you are allowed to be a bit arrogant about it.

(down to #8 now, I suppose)

Not so much "outside his skill class" as "unlikely to be played by a human". Computers have a really good idea of what the optimal move is, and it's sometimes not a move any human being would play. The main way to detect cheating is to figure out that they're playing closer to the computer's line than anybody without computer assistance.

In this case, his off-the-board behavior, where he seems unable to explain some of the moves he made, is also suspicious. His skill set is such that there's no move he's not capable of playing, though he should at least be able to tell you what he was thinking when he played it.

Move in a critical moment, speed of decision for said move, cheating at this level is probably extremely hard to do it alone which already creates space for leaks...

Proving a positive result is much easier than a negative. A positive “this move right here is not reasonable without a chess computer suggestion ♟” vs “we didn’t find anything suspicious”.

And if you’re cheating, adding another layer of “try to find moves that aren’t suspicious” would be theoretically possible.

That isn't really proving a positive though - maybe the player just got lucky. There aren't that many moves in any given turn, sometimes even a novice will make an inspired move.

True - but identifying a particular move gives a discussion point at least.

The other factor is that Regan's process is designed to not produce false positives - if it says you are cheating, you are almost guaranteed to be cheating.

Exactly - the lack of evidence isn't evidence of lack; a common mistake.

All we know now is that an "obvious" method of cheating wasn't used. That may be because there was no cheating at all, or a new method of cheating has been developed.

The fact that Caruana has not said why he has "absolute certainty" the person cheated is cause to be skeptical of Caruana's skepticism.