There is a Japanese proverb "Every stone lost in a ladder is worth 7 points."

I expect the value 7 (shichi) comes from alliteration with shicho (the proverb is: shicho no nobidashi shichimoku no son).

But it made me wonder whether AI can give such a more accurate value.

A constant value makes no sense because environments differ.

KataGo certainly can give you points lost (or gained!) for each stone player out of a ladder. lightvector posted some interesting ladder analyses recently where playing out the ladder with correct timing as a probe can actually be beneficial to amplify the cost of ignoring the ladder breaker (like Lee Sedol's famous ladder game).

But for the simpler case the provern relates to of running out a working ladder that's just a dumb mistake, here's some numbers from a shallow KG on my phone from the first simple ladder joseki that popped into my head.

A 5-4 joseki in lower right where white has the ladder. In this position KG reckons black is already 2 points ahead or 64% and best move is c16 because the ladder threat shackles white's responses.

Click Here To Show Diagram Code

[go]$$Wc
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . M O . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
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$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . 7 . O . . . |
$$ | . . . X . . . . . , . 9 4 1 . , . . . |
$$ | . . . . . . . . . . . . 3 2 5 X . . . |
$$ | . . . . . . . . . . . . . 6 8 . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Black runs out once now white is 73% or +2.5, so black lost 4.5 points.

Click Here To Show Diagram Code

[go]$$Bc
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
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$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . 1 O . O . . . |
$$ | . . . X . . . . . , . O X O . , . . . |
$$ | . . . . . . . . . . . . O X O X . . . |
$$ | . . . . . . . . . . . . . X X . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Black runs out second time, now white is +12.5, so diff of 10.

Click Here To Show Diagram Code

[go]$$Bc
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . 2 . . . . . . |
$$ | . . . . . . . . . . . 3 1 O . O . . . |
$$ | . . . X . . . . . , . O X O . , . . . |
$$ | . . . . . . . . . . . . O X O X . . . |
$$ | . . . . . . . . . . . . . X X . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

3rd time, white 17 so diff of 4.5
4th time, white 21 so diff of 4
5th time, white 26 so diff of 5
6th time, white 31 so diff of 5
7th time, white 37 so diff of 6

So the first one is not so bad, second is big bad, and then more like 5 rather than 7 points per stone fairly stable. In this example at least.

Repeating this for a variety of positions could give some more general estimate.

Click Here To Show Diagram Code

[go]$$Bc
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . O . . . . . |
$$ | . . . O . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . 8 B . . , . . . . . , . . . |
$$ | . . . . 4 7 9 0 . . . . . . . . . . . |
$$ | . . . O 3 5 6 . . . . . . . . . . . . |
$$ | . . O X 1 2 . . . . . . . . . . . . . |
$$ | . . X O O . . . . . . . . . . . . . . |
$$ | . . X . . . . . . . . . . . . . . . . |
$$ | . . X O . . . . . , . . . . . X . . . |
$$ | . . . . . X . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ --------------------------------------[/go]

My 15-block Katago says that

loses 7.7 points
loses 7.4 points
loses 6.2 points
loses 6.2 points
loses 5.4 points
(the marked stone) loses 6.3 points

(Conversely, if the stone O17 was at R14, then each white move would lose roughly 3 points.)