Life In 19x19

Yep, you mostly got it.

But in an actual game, where the number of moves may or may not be even, a komi of 6 shouldn't be the same in practice as a komi of 5.

No, us stopping to do the accounting at the point where the number of moves is even has nothing to do with the game - it's purely a choice of convention for us, so we have fewer cases to consider when doing the accounting. If the number of moves is not even in an actual game, i.e. black has made one more move than white at the point where the last territorially-valuable move is taken and only dame are left, then let white fill one dame before doing the accounting. The again the exact same conclusions hold - komi 6 is the same as komi 5 unless you have one of the rare special cases (odd-dame seki, etc).

If you don't like that convention, then you can feel free to do accounting directly with black having made one more move than white. Suppose we stop the game at that point, where black is ahead by 6 points on territory, and black has made one more move than white. Then, it must be the case that the number of dame is even, so under area scoring black will get the last dame, ending with having made one more move than white, and again will win by 7 points before komi. On the other hand, if black is ahead by 5 points on territory, and black has made one more move than white, then it must be the case that the number of dame is odd, so under area scoring white will get the last dame, catching to be equal in moves with black, and so black will win by 5 points before komi.

You can see that this case is exactly the same as the other one, except with the reasoning about "odd" and "even" reversed, so that it still comes out that komi 6 and komi 5 are equivalent. The point of stopping the game at a point of equal moves is so that we can not have to do the same reasoning twice with flipped odd and even, it doesn't stop the reasoning from applying to actual games.

Also, all of the above reasoning about komi 5 and komi 6 being the same *also* applies regardless of perfect or imperfect play. The only point where imperfect play comes is in order to deduce that "even" and "odd" are roughly equally likely, it plays no role in the above reasoning otherwise. Obviously with perfect play, it may be the case that 100% of games are "odd" dame, or 100% of games are "even" dame, if the players are always playing the same perfect games over and over, instead of being about 50-50. But the above odd/even reasoning will still apply to each game and make komi 5 = komi 6, so long as that perfect play doesn't involve any odd-dame seki or things like that.

If you still find all this confusing and feel like you're likely to forget some of these details again or mix up which things depend on which assumptions, you can put in the time yourself to work it out. You can get a 7x7 board where it's easy to count the score because the board is small and go through example after example, until you find it absolutely intuitive for yourself why:

* The area score difference is almost always odd on a 7x7 board (so that komi usually only matters in increments of 2)
* The dame by itself can easily be either even or odd
* The players' number of moves difference by itself can easily be either be even or odd
* The territory difference at the end by itself can easily either be even or odd
* The individual even/oddness of these things - dame, moves difference, territory difference - *combines and cancels* in a way that is consistent with the final area score difference almost always being odd despite the fact that each one individually could go either way.

(and the above is true on odd sized boards. It comes to be even instead on even sized boards).

lightvector wrote:you can put in the time yourself to work it out. You can get a 7x7 board

This is, of course, not needed any more. Besides Theorems 140 and 143, see

https://www.lifein19x19.com/viewtopic.p ... 43#p277643
https://www.lifein19x19.com/viewtopic.p ... 32#p277832

I have also proved the following on some two pages:

"
Presuppositions

Assume no handicap stones and single passes. Apply the territory score together with the conditions 'unequivocal', 'without two-sided dame and teire', 'without capturable stones in sekis', 'without asymmetric sekis'. Have standard area komi.

Theorem 141 [relation between area and territory scoring]

If Black and White make an equal number of plays so that White makes the last play, the scores are the same under area or territory scoring.

If Black makes one more play than White so that Black makes the last play, the area score is 1 larger than the territory score.

Remarks

The presuppositions do not need board parity and seki parity. The last play might be the last dame. [...]
" [28]

https://www.lifein19x19.com/viewtopic.p ... 45#p143245