I’m guessing this is green:
And these are red:
(All the green positions so far look fairly balanced in some sense. Testing for same number of black and white stones on the nth line. 598 tests for if only the sum of the height of the stones must be balanced. )
Ok, I’ll guess the rule: The sum of the heights of black and white stones must be the same. So three stones on the first line balance one stone on third line
The rule-guess is maybe a little ambiguous? In my red counterexample, the northeasternmost black stone is supposed to be on the 2nd line, but in another sense it’s also on the 3rd line.
EDIT: I just realized this is true of my green counterexample, too. In a sense, the black stone is on the fourth line, satisfying your rule.
Less ambiguous counterexamples:
I suppose you meant that the “height” of a stone is the lowest line it occupies, in which sense my first two counterexamples also work. 
I said at the outset that my rule was defined for all board sizes. That was under the assumption that there were only 5 x 5, 7 x 7 and 9 x 9 boards. Browsing through this thread, I’ve realized that arbitrary board sizes are possible.
My rule is only defined for rectangular boards.
So it works on any n x m board, a 4x6 board would also be allowed?
Yes. After further contemplation, the rule is defined for rectangular boards with at least side length 2.
((Sneaky late EDIT: Technically maybe I should have said side length 1? I meant “2 intersections”, which I suppose could be said to be “1 unit of distance” in a sense.))
The rule is written on this piece of paper. The rule is forty words long, including a nine-word provision that I think makes the rule easier to guess.
If at any point after the rule is discovered, I can be shown to have miscolourised a koan, I will publicly eat the piece of paper.
You may vote (for 24 hours) to:
0 voters
So far, the provision would not have changed the colour of any koan.
Ok what we know so far is that shifting the stones by one can change color of the koan. That means that the rule makes reference to some absolute board position. Since the rule is symmetric under reflections and rotations and can be generalized to any rectangular board, I suspect that it’s either somehow about the edge of the board (e.g. distance from the edge is important) or the center of the board.
Furthermore all the koans with a stone at the center were red and all koans, where the white stones are just a reflection/rotation of the black stones were green, but it’s not a necessary condition. Those could be coincidences, so I’ll try to falsify with the following koans:
The poll has closed, with its three participants voting “I’m indifferent”. You will now enter expert mode and guess the (IMO harder) 31-word version of the rule, since LKSFRZ seems on track to solving it solo. (As before, no koans would have been coloured differently so far.)
Aside from 6 koans by myself and 1 by hoctaph, all koans so far have been supplied by LKSFRZ… Rise from thy slumbers, OGF, rise!
I’ve been a bit busy these last days, but I haven’t forgotten! Here’s my input.
It seems that the hypothetical rule “The sum of pyramid values is the same for Black and White” works for almost all revealed koans so far, but not quite. In particular, 604 is a counterexample:
But I do think that the rule might be similar. In other words, maybe the distance of stones / chains from the center / edge is important.
Should we assume that it’s something like “property of the black stones = property of the white stones”, where the property has something to do with absolute position and try to figure out that property, or should we try to challenge those assumptions a little. E.g. trying to find a green koan with only one color of stones, or a red koan that is symmetric (although 616 makes me quite confident in that assumption).
If we try to figure out the property, then I think finding a red koan similar to 604, 606 and 617 would be most helpful, since they don’t have a lot of symmetry and yet are quite simple
[Post goes “poof”. What you thought you saw here? It never happened. Except maybe in your imagination. How would I know?]
Regardless of how this plays out, after we have this puzzle finished, I would like to see your specific wordings to understand the distinction.
I find the patterns across this set of koans to be quite interesting
