That’s pretty much ruled out by the confirmation that the rule cares only about the graph as it is presented 
The torus would be a different graph.
The answer is:
show
Yes the rule cares about the number of stones in some way.
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How about interpolating between these two?
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These are both red:
Red too.
On a board with one black stone, how many white stones are hinreichend to turn the board green?
It depends on the placement of the black stone, so answering your question is difficult. However maybe this helps a bit:
Also I have prepared a major hint that doesn’t reveal too much and is interesting. If you want to hear it, just tell me.
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Do you know this feeling when you’re carrying your umbrella against the wind and the rain, and you just see the road but not what’s in front of you? Or when you’re forced to bow your head down and can’t look up. That’s how I’m feeling about the puzzle. I know it’s got something to do with liberties and direct contact, but can’t say what and I feel like I’m not even able to find the solution without making babysteps like this one:
If there is only one black and one white stone, the white stone must touch the black stone; but even that is not enough, as seen here: 
But this is strange too:
What is the difference? On the red one, all stones have four liberties, on the green one, they have three. But because we know now that edges and lines don’t matter, it must be connected to the exact number of liberties.
This makes me wonder another thing: Does the Rule contain a specific number?
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Don’t they have two in this example?
the answer is ...
no
yes, of course. But what if? 
… then it becomes red
Those last two boards are indeed very interesting, especially given the above answer.
(seems like @RubyMineshaft has added a cool new feature to give koans numbers so that anyone can load them quickly!)
And this?
Mixed results!
This satisfies the rule!
What about this then?
The games have started to run together in my head. Sorry if this breaks some aspect of the rule we’ve already figured out.
I wish I could say I figured out some part of the rule 
If you jump back to this post, it has some spoiler tags that were given/confirmed.
I also definitely have no idea whether that koan is green or red 
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For a brief second, I was wondering if the number of stones must conform to some weird relation like 8/13 or other fibonaccinanigans. I hope I’m wrong
And can we do a one-dimensional koan? Say, on a 1x5-Board, what is this?
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