As this is currently a one-to-one dialogue, we could go back to email
But, assuming someone else is interested, here are some further thoughts prompted by your reply.
I think you are now seeing (very quickly) that the thrust of SEA is to identify Z(P) rather than to follow-the dual-moves to a shallower depth than P's. The problem with the latter is White can keep conceding depth, and diving deeper. This all gets very messy, and eventually I gave up on this approach entirely.
I wondered whether it was aesthetically right to set the current position P's value to 'draw', or whether 'broken' or 'unknown' would be more appropriate.
We certainly create 'unknown' values temporarily, and the rules of retro'ing back have to include retro'ing back an unknown-value. So, yes, we could set P to unknown initially - but should we? If we do, then all the btm positions that move to P (which could have been set to 'draw') would be set to unknown too. So there would be more 'unknown' positions at the end of the 'killing-off 1-0s phase' than need be, and therefore I think more tidying up to do in phase 2.
'Unknown' has to mean "draw or 1-0" in the algorithm and therefore doesn't reflect the fact that we are pretending that P is _not_ a 1-0 win for White.
So we should set P to 'draw' or 'broken': by the way, setting P to 0-1 just creates more perturbation than we need. Choosing 'broken' as the new value is more honest, but choosing 'draw' is more pragmatic in that it probably gives a more efficient algorithm - and it does provide the perturbation to EGT E that we need.
I'm happy with the economy of Z(P) rather than Z({P}) etc.
Certainly, Z(P) and Z(P') might overlap. I used to think of Z(P) as a 'Basin of Attraction' B(P), borrowing the mathematical concept, but there are some problems in doing that. The Z(P) are not disjoint basins, and P is not a point of stability. The concept of 'attraction' works as well here as in the mathematical basins, which is to say - not that well.
However, I think your claim that overlapping Z(P) and Z(P') means two finales for a study is based on an assumption that White's moves in Z(P) are unique. This is not so, and the Saavedra study is our ilustration as before. White must get its K from Kb4 to Kc2 and has the choice Kb3-Kc2 or Kc3-Kc2, so - strictly speaking - no 'unique winning moves' there.
However, if you say that the theme of the study is to get from somewhere in Z(P) to P, then these 'reconverging lines' may artistically be seen as ignorable - both being entirely in Z(P) - and maybe there's a computer algorithm here too.
I think we need to work in terms of specific examples, to check the conceptual arguments here. As I suggested, try implementing SEA by hand in the context of yr KQRKQR study.
email preferred for me; we can always come back to the board with an attached file tidying all this up.
g