jann wrote:
Your rules do not seem to give the expected (Japanese-like) result in the revised J89 example in question (bent4 + combined double ko) as that is a case where your two patches can not complement each other (like they do in combined moonshine cases).
OK, I will provide you with the "solution" earlier than intended, but which Gérard will not like at all, as it will be proven that GT City would have better opened its gates to dogs, especially Labradors like Honte with his "magical DOUBLE-ko spell".
++++++++++++++++++++++++++
- Click Here To Show Diagram Code
[go]$$W
$$ +----------------------
$$ | . O . X O X O . . . .
$$ | O X X X O X O O . . .
$$ | . X O O O X X O O . .
$$ | X X O O . O X X O , .
$$ | O O O . O X . X O . .
$$ | X X O O X . X X O . .
$$ | . X X O O X X O O . .
$$ | X . X X X O O O . . .
$$ | X X X O O O . . . . .
$$ | O O O O . . . . . , .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .[/go]
Black claims "controlled area" in the entire upper left corner.
Black is the "defender"; White is the "attacker", moving first.
- Click Here To Show Diagram Code
[go]$$W
pass
$$ +----------------------
$$ | 1 O . X O X O . . . .
$$ | O X X X O X O O . . .
$$ | . X O O O X X O O . .
$$ | X X O O 2 O X X O , .
$$ | O O O . O X . X O . .
$$ | X X O O X 3 X X O . .
$$ | . X X O O X X O O . .
$$ | X . X X X O O O . . .
$$ | X X X O O O . . . . .
$$ | O O O O . . . . . , .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .[/go]
The defender's pass with

creates a "critical position", as the attacker captured a ko with

just before. Black claims for a "permanently prohibited ko", i.e. White's capture into the lower ko-shape of the double-ko is prohibited next time.
Please note that this will make the double-ko disappear AFTER the next ko-capture in it (Honte is lurking around the corner...).
- Click Here To Show Diagram Code
[go]$$W
$$ +----------------------
$$ | O O 5 X O X O . . . .
$$ | O X X X O X O O . . .
$$ | 6 X O O O X X O O . .
$$ | X X O O X . X X O , .
$$ | O O O . O X . X O . .
$$ | X X O O . O X X O . .
$$ | . X X O O X X O O . .
$$ | X . X X X O O O . . .
$$ | X X X O O O . . . . .
$$ | O O O O . . . . . , .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .[/go]
- Click Here To Show Diagram Code
[go]$$W
pass
$$ +----------------------
$$ | . 7 . X O X O . . . .
$$ | . X X X O X O O . . .
$$ | X X O O O X X O O . .
$$ | X X O O X 9 X X O , .
$$ | O O O . O X . X O . .
$$ | X X O O 8 O X X O . .
$$ | . X X O O X X O O . .
$$ | X . X X X O O O . . .
$$ | X X X O O O . . . . .
$$ | O O O O . . . . . , .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .[/go]
The defender's pass with

creates a "critical position", as the attacker captured a ko with

just before. Claiming for a "permanently prohibited ko" is irrelevant here.
- Click Here To Show Diagram Code
[go]$$Wm11
pass
$$ +----------------------
$$ | 2 O . X O X O . . . .
$$ | 1 X X X O X O O . . .
$$ | X X O O O X X O O . .
$$ | X X O O . O X X O , .
$$ | O O O . O X . X O . .
$$ | X X O O X x X X O . .
$$ | . X X O O X X O O . .
$$ | X . X X X O O O . . .
$$ | X X X O O O . . . . .
$$ | O O O O . . . . . , .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .[/go]

gives atari in the corner.
Playing at

, instead, will not lead to success either.

can only pass, as "x" is a "permanent prohibited ko" for her, as attacker.
- Click Here To Show Diagram Code
[go]$$Wm11
$$ +----------------------
$$ | X O a X O X O . . . .
$$ | 5 X X X O X O O . . .
$$ | X X O O O X X O O . .
$$ | X X O O 4 O X X O , .
$$ | O O O 6 O X . X O . .
$$ | X X O O X x X X O . .
$$ | . X X O O X X O O . .
$$ | X . X X X O O O . . .
$$ | X X X O O O . . . . .
$$ | O O O O . . . . . , .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .[/go]

captures into the double-ko here, also giving atari at White's group therein.
As a matter of course, he could also succeed by capturing at "a" in the corner first.

can only re-capture in the corner, as a move at "x" is prohibited for her.

captures White's double-ko group.
- Click Here To Show Diagram Code
[go]$$Wm11
$$ +----------------------
$$ | . O . X . X O . . . .
$$ | O X X X . X O O . . .
$$ | X X . . . X X O O . .
$$ | X X . . . . X X O , .
$$ | . . . . . X . X O . .
$$ | X X . . X . X X O . .
$$ | . X X . . X X O O . .
$$ | X . X X X O O O . . .
$$ | X X X O O O . . . . .
$$ | O O O O . . . . . , .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .[/go]
The claimed "controlled area" is Black territory!!!
You will easily realise that this does NOT match J89's expectations!
++++++++++++++++++++++++++What are the reasons for this striking result?

Well, you will have realised that the DEFENDER (sufficient liberties of their attachment to the double-ko provided) will be able to kill the attacker's double-ko group.

L&D Example 18 contains a sandwich with FOUR layers (i.e. even). The DEFENDER owns the OUTER double-ko shape (but which is the one expected to die).

The ATTACKER's layer at the other end of the sandwhich (here in the corner) is NOT CONNECTED to the OUTSIDE. This implies that BOTH layers of the ATTACKER will die in the process.
What is the difference to L&D Example 16 (double-ko with attached moonshine-life).

The attacker's double-ko group cannot be killed, due to the liberty-count of the attachment.
(In the case of nested ko-shapes for the inner double-ko group, the defender's double-ko group could survive. However, this would not help her, as she would be unable to find any suitable "controlled area".)
L&D Example 16 contains a sandwich with THREE layers (i.e. odd).

The DEFENDER's layer at the other end of the sandwich is CONNECTED to the OUTSIDE (i.e. Black's encirclement). The DEFENDER will be unable to prevent the ENTIRE attachment from becoming captured. Thus, there will be NO territory for the DEFENDER in any claimed "controlled area".
(This would be true even in the case she captured the attacker's double-ko group.)++++++++++++++++++++++++++By the way:
The same principle would apply here, in a corner position WITHOUT ko, which is probably easier to understand.
- Click Here To Show Diagram Code
[go]$$B
$$ +----------------------
$$ | O . X O O O X X O . .
$$ | O O X O O . O X O . .
$$ | . . X O . O X X O . .
$$ | X X X O O X . X O , .
$$ | O O O O X . X X O . .
$$ | X X X O O X X O O . .
$$ | . X . X X O O O . . .
$$ | X X X X O O . . . . .
$$ | O O O O O . . . . . .
$$ | . . . , . . . . . , .
$$ | . . . . . . . . . . .[/go]
As a matter of course, White could also claim the corner being her "controlled area", and succeed.
White has 19 stones in the corner, Black has 18 stones. Thus, the corner is worth a surplus of two points of territory for Black.