With no kos or sekis, chilling is equivalent to cooling by 1 pt. Cooling does not affect the count (mean value). This was proved in On Numbers and Games.



As for kos and superkos, there is no single way of evaluating them. They are not combinatorial games. That is why we talk about mast values rather than mean values.

You know that territory scoring may be regarded as chilled area scoring. And, because of rule differences regarding kos and sekis, those scores may differ by more than one point. (Infinitesimals in territory scoring, normally just dame, can make a difference of one point between the territory score and the area score, just as infinitesimals in chilled go can make a difference of one point in territory scoring.)

As far as I know, nobody actually uses chilling to study ko positions.



Under area scoring each board play adds one point to the player's territory. Territory scoring chills area scoring by not adding that point for each board play. Each board play in chilled go subtracts one point from the player's territory.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Very nice!

This means that my last two hours have been wasted, except for having performed exercises, with which I only prove special cases:


Theorem:

The count C of the local gote {R||S|W} (with R>S>W) equals the count C' of the chilled game {R-1||S|W+2}.

Proof:

As a local gote, {R||S|W} has the count C = (R + (S+W)/2) / 2.

The count of the white follower {S|W+2} of the chilled game is 1 larger than the count X := (S+W)/2 of {S|W} of the unchilled game. Therefore, the gote count of the chilled game is the gote count of {R-1|X+1} and is C' = ((R-1) + (X+1)) / 2 = (R+X)/2 = (R + (S+W)/2) / 2 = C. QED.


Theorem:

The count C of the local sente {R||S|W} (with R>S>W) equals the count C' of the chilled game {R-1||S|W+2}.

Proof:

As a local sente, {R||S|W} has the count C = S.

Lemma: The chilled game {R-1||S|W+2} is a local sente.

Proof of lemma:

Using the assumption of having a local sente, we apply that a local sente fulfils R - S < (S+W)/2 (the sente move value is smaller than the follow-up move value).
R - S < (S+W)/2 =>
(R - S) - 1 < (S+W)/2 + 1 =>
R - 1 - S < (S + W + 2) / 2 =>
(R-1) - S < (S + (W+2))/2 =>
we can substitute R' = R-1 and W' = W+2, apply the assumption again so the chilled game {R-1||S|W+2} also is a local sente. QED.

As a local sente, its count is C' = S.

We have C' = C = S. QED.

QUESTIONS 11:

This is about Figure 2.12 and its procedure in chapter 2.5 in conjunction with Theorem 5 in chapter 4.4.

What does "pair off infinitesimals which are negatives of one another and therefore add to 0" mean?

The theorem speaks of "no two summing to zero" so I suppose it only applies pairwise. Pair off

- star with star
- up-x with down-x
- up-x-star with down-x-star
- tiny-x with miny-x
- 0^n|tiny-x with miny-x|0^n

Is it correct that the procedure and theorem do NOT mean to ignore any combination of at least three infinitesimal values summing to zero? E.g., is it correct that we are NOT supposed to ignore the combination of the values up-2-star + down + down + star, which sum to zero but are not a value pair (they are not equal options aka miai, as we call it)?

Is it correct NOT to pair off tiny-x with miny-x|0^n?

Is it correct NOT to pair off miny-x with 0^n|tiny-x?

Is it correct NOT to pair off 0^n|tiny-x with miny-x|0^m; n <> m?

What exactly does "attack long corridors and/or defend attacks on tinys" mean? What corridors are "long"? Whose tinys? What is a defense against an attack on a tiny?

What does "The remaining infinitesimals should all be positive [...]" mean, considering that the previous step of the precedure included "attack long corridors and/or defend attacks on tinys"? What long corridors are "positive", are tinys "positive" and in which sense is either "positive"?

Does Figure 2.12 and its move order only apply to the chilled game or does it equally apply to normal unchilled go and its move order? Does the procedure only apply to the chilled game or does it equally apply to normal unchilled go?

EDITS

Assuming that "off" is a typo for "of", it means what it says. X + (-X) = 0.



I'll take a look in the morning, so I am not sure. OC, it is possible to have three infinitesimals that add to 0. E. g., ^* + v + * = 0. If you have any games that add to 0 and there are no kos, offhand I don't know why you wouldn't ignore them.





Yes to all three.



I guess they are taking Black's point of view. A long corridor I expect is one with a non-zero atomic weight. As Black you want to eliminate negative positions so that White does not get the last play.



Black's, I suppose, since tinies are positive.



Answer the sente.



An infinitesimal is positive if Black can get the last local play, no matter who plays first.



The book uses chilling to find correct play in the unchilled game. Unless there are kos, correct play in the chilled game is also correct in the unchilled game.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Figure 2.12 and Theorem 5 restrict themselves to infinitesimals that are so simple that an odd number of them cannot add to 0. For instance, although you can construct ^* as a single game on the go board, not as the sum of ^ and *, such infinitesimals do not appear in Figure 2.12, and Theorem 5 is even more restrictive.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Asked differently for an example, if we have ^2* + v + v + * + X, can X be on the board so that we are required to ignore ^2* + v + v + * = 0 and start playing in X because any other move, in ^2* + v + v + *, would be a mistake?

We are not required to ignore it, but we can ignore it. It makes our job easier. A move in ^2* + v + v + * might be a mistake. Why take the chance?

BTW, there are a number of problems where there is a miai and playing in the miai would be a mistake. Such positions occur in actual play, as well.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Seems I need to find an example myself...

Click Here To Show Diagram Code

[go]$$W Miai plus miny, White to play in a corridor
$$ -------------------------------
$$ . X W . . . O B . . X . . B B O .
$$ . X X X X X O O O O X O O O O O .
$$ . . . X W . O B . . X . . . . . .
$$ . . . X X X O O O O X . X . . . .
$$ . . . . . . . . . . . . . . . . .[/go]

Unmarked stones are invulnerable.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Very nice example for the requirement to ignore 0 sums! I have tried a fraction, *, up or down but your miny does it! The unfortunate consequence is that infinitesimals must not be avoided even for seemingly simple considerations related to equal options.

Click Here To Show Diagram Code

[go]$$W Correct, smaller count = -5
$$ -------------------------------
$$ . X W . . . O B . . X 1 . B B O .
$$ . X X X X X O O O O X O O O O O .
$$ . . . X W . O B . . X . . . . . .
$$ . . . X X X O O O O X . X . . . .
$$ . . . . . . . . . . . . . . . . .[/go]

The remaining local endgames sum to 0. It is correct for White to ignore them, starting elsewhere in the miny.

Click Here To Show Diagram Code

[go]$$W Mistake, larger count = -4
$$ -------------------------------
$$ . X W . . 1 O B . . X 2 3 B B O .
$$ . X X X X X O O O O X O O O O O .
$$ . . . X W . O B . 4 X . . . . . .
$$ . . . X X X O O O O X . X . . . .
$$ . . . . . . . . . . . . . . . . .[/go]

The remaining local endgames sum to 0. It is wrong for White to attack the local endgames whose initial sum was 0.

Click Here To Show Diagram Code

[go]$$W Miai plus down, White to play in a corridor
$$ -------------------------------
$$ . X W . O O O . . X . . B O . .
$$ . X X X X X O . . X O O O O O .
$$ . . . X W . O . . X . . . . . .
$$ . . . X X X O . . X . X . . . .
$$ . . . . . . . . . . . . . . . . .[/go]

Here is an example with White's only correct play in the down.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

QUESTIONS 12:

What is an example for attacking MINY-x|0^n being better than attacking MINY-x|0^(n+1)?

Click Here To Show Diagram Code

[go]$$B MINY-2|0 and MINY-2|0^2
$$ . . . . . . . . .
$$ . O O O O O O . .
$$ X . . . X X O . .
$$ . O O O O O O . .
$$ . O O O O O O O .
$$ X . . . . X X O .
$$ . O O O O O O O .
$$ . . . . . . . . .[/go]

What is an example for attacking MINY-x|0^n being better than attacking a DOWN-d or DOWN-d-STAR corridor?

Click Here To Show Diagram Code

[go]$$B MINY-2|0 and DOWN
$$ . . . . . . . . .
$$ . O O O O O O . .
$$ X . . . X X O . .
$$ . O O O O O O . .
$$ . O O O O . . . .
$$ X . . X O . . . .
$$ . O O O O . . . .
$$ . . . . . . . . .[/go]

EDIT

White 1 produces a position where Black needs to attack the shorter corridor. Attacking the longer corridor lets White win.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Again, after White 1 Black must not attack the down.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Ah, the difference game, very nice!

QUESTIONS 13:

Mathematical Go Endgames creates the impression of positional judgement being possible by rounding and explains it for numbers, ups, downs, tinys, minys by comparing them to each other.

How to round if also 0^n|TINY-x or MINY-x|0^n are involved?

How do 0^n|TINY-x and TINY-x compare?

How do 0^n|TINY-x and TINY-y compare (x<>y)?

How do 0^n|TINY-x and 0^n|TINY-y compare (x<>y)?

How do 0^m|TINY-x and 0^n|TINY-x compare (m<>n)?

How do 0^m|TINY-x and 0^n|TINY-y compare (m<>n, x<>y)?

What is the white attacker's incentive in 0^n|TINY-x (n>1)?

What is the white attacker's incentive in 0|TINY-x?

Why is 0^n|TINY-x about UP-n + STAR^n? Slightly smaller or larger? What does this tell us for rounding?

How do uptimals compare to ordinary infinitesimals?

Without answering these questions, positional judgement by rounding is a myth, unless there are only numbers, ups/dows, stars, tinys/minys.

For the theory of atomic weights see On Number and Games and Winning Ways.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

QUESTIONS 14:

These questions are about a group invading several corridors.

Has Mathematical Go Endgames only studied empty corridors?

Where to play as attacker / defender when one string with one socket invades blocked and / or unblocked empty corridors?

Where to play as attacker / defender when one group with multiple sockets invades blocked and / or unblocked empty corridors?

Information on this is encrypted in the proofs and I could not decipher it yet.