I agree, this is the first annoying question to ask. As a seasoned club player, i believe to know that "clean Pawn up" means chess wise roughly what you think it means, namely: a) You are in a position, where it really does make sense to count pawns, a so called "quiet" position would be one of those. b) Your opponent has no pawn compensation of any kind, not even in the form of a nuisance. I personally learned this expression from GM Nakamura. Clean pawns were unknown before the "golden era of chess", you could utilize the idiom "ceteris paribus" instead, this was used by the old german vice WM Tarrasch.

Hence it should be clear to anyone that without engine assistance it is totally impossible to determine for 10^6 games if a certain material difference was "clean" or not. Most likely he just checked for a stability over n consecutive moves, at least this is the usual way. Others have noted more problems, that might help put his highly deviating, washed out results in order.

Contrary to the definitions in the other answers, I'd assume that this refers to selecting positions from the database where that's the only difference in material - so "clean up a pawn" means that all the other figures are the same, that there isn't also a bishop vs knight difference, for example.

I really doubt if there was any attempt to check if there's some "strategic compensation" - how would you do that at a large scale? I doubt that even running a solid chess engine evaluation on all these positions is feasible, you need something where you can simply/cheaply filter positions from the database and then just count the winrate.

Apart from what WJW said, it's not an objective criterion but it essentially means that you don't lose out in terms of initiative or position. A counterexample might be accepting the queen's gambit with black (i.e. 1. d4d5 2. c4cxd4), that way you win a pawn but you arguably give white an advantage in development (an in many cases white will be able to recapture the pawn as a result)

"Cleanly" in chess parlance means you're up a pawn, and there is no "compensation" for the other player. I.e there are no other advantages in the opponents position that make up for the fact they're down a pawn(things like an attack on your king, piece activity, pawn structure).

In simple mathematical terms a clean material advantage is:

Overall eval >= material eval

If overall eval < material eval, then compensation(other player) > 0 = "the opponent has some compensation for the pawn"

I agree with this question. How was clean determined in the dataset?

Knowing how the word is used in chess is not enough to know where the line was drawn in the data here, which is the basis for the entire analysis shown.

From the other imbalances listed (like being up "the exchange"), I take it that they mean being up a pawn without having to give up anything in return. For pawns that is kinda obvious, but for more valuable pieces like a rook you can win it "cleanly" or maybe you take it with a pawn and then immediately lose the pawn.

You can of course give up material for positional advantages as well, but from the article I don't think the author analyzed that. It would be difficult to accurately measure that anyway, some positions can be up +5 points according to the computer but only if you find 20 perfect moves in a row. Needless to say, most humans would not find those moves especially in the lower ELO brackets that the article analyzed.

Should have been defined (along with more information about the data set), but I assume it means that the opponent has no tactic that wins them material. For instance if I just take their protected bishop with my bishop then I'm briefly up a bishop. So it would not include that situation or where the trade is more complicated.

I think it means that the opponent does not gain a strategic advantage from an exchange. You both have good enough positions and you are simply a pawn up.