Gérard TAILLE wrote:
- Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | W X . B M O . |
$$ | . W B B B O O |
$$ | W . W W B B O |
$$ | W W W . W B O |
$$ | X X W W B B O |
$$ | . X X W W W . |
$$ | X . X X X W W |
$$ -----------------[/go]
Because of the
external marked liberty you can no more use magic wand can you?
- Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | O X . B C W . |
$$ | . W B B B W W |
$$ | W Q W W B B W |
$$ | W W W . O B W |
$$ | X X W W B B W |
$$ | . X X W W W Q |
$$ | X . X X X W W |
$$ -----------------[/go]

is a SHARED INTERNAL liberty of Black's and White's (only single;

added for clarity) groups, isn't it?

++++++++++++++++++++++++++++
Nevertheless, I can understand your concerns about a combination of two ko-shapes, which are caught in a (potential) semeai.
So, let's return to the "standard" version of a double-ko...
- Click Here To Show Diagram Code
[go]$$B
$$ +---------------+
$$ | . Y . Y O X . |
$$ | X X Y O O X . |
$$ | Q X O . O X . |
$$ | . Q O O O X . |
$$ | Q O X X X X . |
$$ | X X X . . . . |
$$ | . . . . . . . |
$$ +---------------+[/go]
Double-ko, standard version.
- Click Here To Show Diagram Code
[go]$$B
$$ +---------------+
$$ | C B C X O X . |
$$ | B B O O O X . |
$$ | O B O . O X . |
$$ | . O O O O X . |
$$ | O O X X X X . |
$$ | X X X . . . . |
$$ | . . . . . . . |
$$ +---------------+[/go]
Black's group has TWO liberties

.
- Click Here To Show Diagram Code
[go]$$B
$$ +---------------+
$$ | . X . X W X . |
$$ | X X X W W X . |
$$ | O X W C W X . |
$$ | C W W W W X . |
$$ | W W X X X X . |
$$ | X X X . . . . |
$$ | . . . . . . . |
$$ +---------------+[/go]
White's group has TWO liberties

.
- Click Here To Show Diagram Code
[go]$$B
$$ +---------------+
$$ | . X Y T O X . |
$$ | X X X O O X . |
$$ | T X O . O X . |
$$ | Q O O O O X . |
$$ | O O X X X X . |
$$ | X X X . . . . |
$$ | . . . . . . . |
$$ +---------------+[/go]
After Honte used his magic wand...
- Click Here To Show Diagram Code
[go]$$B
$$ +---------------+
$$ | S X X C O X . |
$$ | X X X O O X . |
$$ | C X O S O X . |
$$ | O O O O O X . |
$$ | O O X X X X . |
$$ | X X X . . . . |
$$ | . . . . . . . |
$$ +---------------+[/go]
There are two SHARED liberties

now, and each of the double-ko groups has another exclusive liberty

inside. Thus, both groups have THREE liberties now, one more than before.
- Click Here To Show Diagram Code
[go]$$B
$$ +---------------+
$$ | S X X C O X . |
$$ | X X X O O X . |
$$ | Q X O S O X . |
$$ | O O O O O X . |
$$ | O O X X X X . |
$$ | X X X . . . . |
$$ | . . . . . . . |
$$ +---------------+[/go]
Probably we might want to add a supplementary effect to Honte's magic spell, which would be equivalent of FILLING ONE of both ko-shapes.
- Click Here To Show Diagram Code
[go]$$B
$$ +---------------+
$$ | S X X Y O X . |
$$ | X X X O O X . |
$$ | C X O S O X . |
$$ | O O O O O X . |
$$ | O O X X X X . |
$$ | X X X . . . . |
$$ | . . . . . . . |
$$ +---------------+[/go]
The position is SYMMETRICAL, so we could ALSO choose the other ko-shape for this purpose.
This reduces the number of SHARED liberties by one, so we are back at TWO liberties for both double-ko groups.
+++++++++++++++++++++++++
Let's try to analyse the application on GT City's Central Park:
- Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | O X . X . O . |
$$ | . O X X X O O |
$$ | O . O O X X O |
$$ | O O O . O X O |
$$ | X X O O X X O |
$$ | . X X O O O . |
$$ | X . X X X O O |
$$ -----------------[/go]
- Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | O T Y X . O . |
$$ | . O X X X O O |
$$ | O . O O X X O |
$$ | O O O Q T X O |
$$ | X X O O X X O |
$$ | . X X O O O . |
$$ | X . X X X O O |
$$ -----------------[/go]
After Honte used his magic wand...
- Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | O Y X X . O . |
$$ | . O X X X O O |
$$ | O . O O X X O |
$$ | O O O O . X O |
$$ | X X O O X X O |
$$ | . X X O O O . |
$$ | X . X X X O O |
$$ -----------------[/go]
Supplymentary effect for the UPPER ko-shape.
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[go]$$B WHITE territory?
$$ -----------------
$$ | O X X X . O . |
$$ | 1 O X X X O O |
$$ | O 2 O O X X O |
$$ | O O O O . X O |
$$ | X X O O X X O |
$$ | . X X O O O . |
$$ | X . X X X O O |
$$ -----------------[/go]
Me ari me nashi. => White territory!
- Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | O . X X . O . |
$$ | . O X X X O O |
$$ | O . O O X X O |
$$ | O O O O Q X O |
$$ | X X O O X X O |
$$ | . X X O O O . |
$$ | X . X X X O O |
$$ -----------------[/go]
Supplymentary effect for the LOWER ko-shape.
- Click Here To Show Diagram Code
[go]$$B WHITE territory?;
pass
$$ -----------------
$$ | O . X X . O . |
$$ | . O X X X O O |
$$ | O . O O X X O |
$$ | O O O O O X O |
$$ | X X O O X X O |
$$ | . X X O O O . |
$$ | X . X X X O O |
$$ -----------------[/go]
Me ari me nashi. => White territory!
It does not wonder that the sequences are different, as the formation is NOT symmetrical.
The results are the same as without supplementary effect. => GENUINE double-ko.
++++++++++++++++++++
- Click Here To Show Diagram Code
[go]$$B
$$ +-------------------+
$$ | O O . X O . O X . |
$$ | O X X X O O X X X |
$$ | X X X X O X . X O |
$$ | O O O . O O X X O |
$$ | . X X X X X O O O |
$$ | X X . X . . O . O |
$$ | . X X X . . O O . |
$$ | X X . . . . . O O |
$$ | . . . . . . . . . |
$$ +-------------------+[/go]
Your example with the pretended double-ko in the upper right.
- Click Here To Show Diagram Code
[go]$$B
$$ +-------------------+
$$ | O O . X O Q T X . |
$$ | O X X X O O X X X |
$$ | X X X X O T Y X O |
$$ | O O O . O O X X O |
$$ | . X X X X X O O O |
$$ | X X . X . . O . O |
$$ | . X X X . . O O . |
$$ | X X . . . . . O O |
$$ | . . . . . . . . . |
$$ +-------------------+[/go]
After Honte used his magic wand...
- Click Here To Show Diagram Code
[go]$$B
$$ +-------------------+
$$ | O O . X O O Q X . |
$$ | O X X X O O X X X |
$$ | X X X X O . X X O |
$$ | O O O . O O X X O |
$$ | . X X X X X O O O |
$$ | X X . X . . O . O |
$$ | . X X X . . O O . |
$$ | X X . . . . . O O |
$$ | . . . . . . . . . |
$$ +-------------------+[/go]
Supplymentary effect.
Please note that the compound of the two ko-shapes is a symmetrical one here, so we are free to choose which one of these we want to fill.
- Click Here To Show Diagram Code
[go]$$W BLACK territory?
$$ +-------------------+
$$ | O O 2 X O O O X . |
$$ | O X X X O O X X X |
$$ | X X X X O . X X O |
$$ | O O O 1 O O X X O |
$$ | . X X X X X O O O |
$$ | X X . X . . O . O |
$$ | . X X X . . O O . |
$$ | X X . . . . . O O |
$$ | . . . . . . . . . |
$$ +-------------------+[/go]
- Click Here To Show Diagram Code
[go]$$W BLACK territory?
$$ +-------------------+
$$ | 3 . X X O O O X . |
$$ | . X X X O O X X X |
$$ | X X X X O . X X O |
$$ | O O O O O O X X O |
$$ | 4 X X X X X O O O |
$$ | X X . X . . O . O |
$$ | . X X X . . O O . |
$$ | X X . . . . . O O |
$$ | . . . . . . . . . |
$$ +-------------------+[/go]
Two liberties for Black vs. only one liberty for White. => Black territory!
This result is different from that without supplementary effect => The supposed seki is NOT a GENUINE one!
Thus -- when in doubt (or too lazy to do a thirteen minute double-ko analysis, including an intensive internet recherche) -- view the director's cut, extended version

I am sure that you will not get tired of this film. You will not stumble over (critical) double-ko formations in EVERY of your games.
_________________
The really most difficult Go problem ever:
https://igohatsuyoron120.de/index.htmIgo Hatsuyōron #120 (really solved by KataGo)