toggling the opposition

i just found the endgame practice section of chess dot com. 

at first this was very exciting because the king and pawn endgame puzzles are very puzzly. its interesting that the very specific king moves can end in either a win or a draw. i could tell there was something to understand, yet i didnt understand it.

sometimes the enemy king would manage to block me entirely, and i couldnt get out of it.

well, that led me down an opposition rabbithole.

i have found all the explanations to be very bad.

i get how having opposition works in the moment, and how to get opposition if ur almost at opposition. but u can lose opposition, or gain opposition, depending on where the pawns are and other factors. 

if theres nothing else on the board, its straightforward. but thats a draw anyway. counting even and odd squares or whatever doesnt tell the full story of whats going on, and the lessons ive seen go over things case by case without really explaining the theory.

ive been trying to figure out how to think of it properly, and so far the best ive come up with is to think of it in terms of both opposition and opposition toggles.

take a position like this, white to move:

this is winning for white. and white will need to use opposition to win. in fact, white already has the initiative for opposition across the c file. but if it takes opposition here, its a draw.

instead, first white has to give black the opportunity to gain opposition. isn’t that bad? 

white to move, black gets the opposition:

fortunately for white in this position, if black takes opposition, white can gain it back thanks to there being space for a pawn move that changes the parity of the opposition. after the pawn move, black is forced to break opposition, giving it back to white.

thats simple enough to conceptualize. black has opposition, but white has an opposition parity toggle.

but what if after kc4, black also moves diagonally, to ke5?

white can get into horizontal opposition. but this opposition is not useful in itself, it will just lead to a draw, which is what black wants. and if white moves in front of the pawn from this distance, black gets opposition in a way that cannot be changed by a pawn move shifting the parity, because the pawn as a parity toggle is blocked.

the white king has no productive moves.

the solution is to move the pawn, which essentially pre-toggles the parity of the vertical opposition.

this also changes the parity of the horizontal opposition thats been going on, giving black opposition in theory. but black is blocked from moving downward, because the pawn controls the square. and while black could keep horizontal opposition parity by moving straight up, that leaves the pawn free to move up and toggle right back.

staying on the wrong file wont help black. so the king tries its best to block the pawn by moving to the pawn’s file, with a diagonal move which changes the parity and gives the opposition back to white, and can only hope white messes up opposition in a way that loses both opposition and the ability to toggle.

without that pawn move, if white had done a parity-preserving move such as going to the left, black’s diagonal move would have given it opposition. which is a draw for white.

of course, if white had gone left black could also have gone for the pawn, which would be a faster draw. its a bad move either way, but useful for understanding the mechanics of opposition parity.

all this requires exploring the position really thoroughly when i feel like it should be a more straightforward computation.

no explanations ive found even try to explain how changing the parity matters and how u might go about doing it. they just say get in front of the pawn, and maybe something about particular ranks and files, and they take for granted that black will do the "ideal" move in response.

i feel frustrated cuz theres clearly a theory here but ive spent hours to even understand it enough to successfully get through the timed king+pawn endgame challenges.

 https://www.chess.com/endgames/pawn/winning-king-and-pawn/challenge 

i can do the challenges now, but i dont feel satisfied with my understanding.

but heres my best understanding as it stands, after banging my head against this almost all day:

opposition prevents ur opponents progress. it doesnt just matter who has the opposition in the moment, it matters who has the ability to toggle the opposition parity. toggles include other pieces movement, and diagonal king movement.

when kings are far apart, the concept of diagonal opposition is basically a calculation of who runs out of diagonal toggles first and loses control of toggling the opposition. when a king flanks, its a diagonal movement that gives the enemy king the initiative to either also toggle diagonally, maintaining ur opposition, or to stay on the same rank/file and gain opposition but in the the opposite axis. but that wont be productive in stopping a pawn if its positioned out of ur control, leaving toggle power to the enemy.

if ur clever u can sometimes use a blocked square to force the enemy on a diagonal, making them toggle the power back in ur hands.

so if ur relying on opposition to draw, u need to control the toggles. if ur going for a win, u need to protect ur pawns and ur toggles.

my brain needs a nap