A previous post by n95 (currently #5637) contained a position with a SUPG of 3, with a longer UPG of 4.5. I'm wondering if this would count as a position with a gap of 1.5, since instead of having a SPG and a longer SUPG, both methods are UPGs.
No. For that position the gap is zero, since the SUPG is also the SPG. For a gap greater than zero the SPG can't be unique.
Obviously the theoretical gap maximum is 1.5 as 2.0 would imply you can play an extra move and undo it.
I'm hoping you can help me understand this. You are saying that a position with a non-unique SPG in 8.5 can't have a shortest unique PG in 10.5 or more, but I'm not sure why that is.
For what it's worth, I know a position with a unique SPG in 8.5 and a unique PG in 13.0. For that position the gap is zero, since the SUPG is the SPG.
My understanding of what Arisktotle said is that either side could move a piece and then move it back to the square it just moved from. Is that correct or completely wrong?
Either way, I would assume that for a gap of 1.5 to be possible, this possibility would need to be stopped, since the side that played the last move has wasted two moves and therefore could have moved a piece and then moved it back to where it started. There's probably many workarounds, but I haven't seen any examples yet. I would love to see a position with a gap of 1.5, if anyone knows of one.
Here's a proof game in 14.0, one of many.
My understanding of what Arisktotle said is that either side could move a piece and then move it back to the square it just moved from. Is that correct or completely wrong?
What position are you referring to from which either side could move a piece?
Well, first definitions. I stuck to my own term "EPG" rather than "SUPG" since it is crystal clear, a proof game in the exact number of moves specified. That it is unique does not follow from the definition but from the fact that "stipulations" in composition chess are commonly met by just one solution. The uniqueness requirement is global - exceptions must always be specified. The unique stretch in endgame study solutions need not include the checkmate since the stipulation is not "checkmate" but "win" which leaves wiggling room.
When I have a proof game solution in 6.5 moves and another one in 8.5 moves, I can always start with 1. Nf3 Nc6 2. Ng1 Nb8 and have it followed by the 6.5 move game to get an 8.5 move proof game. Whatever other proof game in 8.5 I had in mind it is now dualed by the "6.5+2" version + a number of similar extensions. Btw, that is no proof that the lag can't be 2.5 moves but it is easy to see that it is extremely hard to create a proof game that avoids the possibility of longer "zero- or half-move sum" manoeuvres!
It might answer someone's question, but not the one I asked. I will use the term SUPG since it is crystal clear. It's the shortest game that reaches the end position with a unique series of moves. If it is longer than the position's SPG, obviously the SPG must be non-unique. So in trying to make the gap as large as possible, we are only dealing with positions that have a non-unique SPG. I believe your comment that the gap can't be more than 1.5 relates to some task other than the one I proposed at #6002.
Here's a position with a unique SPG in 7.0 and a unique PG in 10.0. Since the SPG is unique, it's also the SUPG, and the gap between the SPG and the SUPG has length zero.
Ah, I see your point. That's why I stuck to my EPG term avoiding the risk of running into a different interpretation of SUPG. Effectively the difference between EPG and SUPG is exactly this:
The EPG stipulation gives you the number of moves of the unique proof game solution while the SUPG asks you to find it. Both ask for that proof game. The difference is very much like "#3" for a directmate problem versus "checkmate-as-fast-as-possible" which after an extra solver effort turns out to be #3 as well.
On the composer level they are precisely the same. The composer needs to design the same thing, the shorter non-unique SPGs, the gap, and the sngle SUPG/EPG. Therefore the answer I gave in my last post is exactly the same. When your candidate unique SUPG proof game is 2.0 moves longer than a set of non-unique proof games then you can always insert 2 "zero-sum" moves in your non-unique SPGs to give them the same length as your SUPG candidate. Which then of course is no longer unique for that solution length!
Btw, In the case that the intended SUPG is unique but the shorter SPG by chance is unique as well, then you can't use the SUPG stipulation but the EPG form still works.
Just see you example and there are some more issues. I would not use SUPG in that way - though it's not my call of course
. Better not to ask for the shortest but for the longest unique series of moves (LUPG). That would generate the same answers because finding unique proof games only gets harder when the move numbers go up. And it would avoid having to add the PG classification in this process!
And of course, the 2.0 zero-sum formula only applies when the SPGs are non-unique as they appeared in the preceding posts. Details matter!
I think I understand that, but am still not sure if your statement at #6003, "Obviously the theoretical gap maximum is 1.5..." is supported by your added explanations.
I don't want to use LUPG because when there are several unique proof games, I care about the move length difference between the SPG and the shortest of them.
The position at #6002 has four SPGs in 3.0, a unique PG in 4.0 (the SUPG), and a unique PG in 4.5 (possibly the LUPG). The move length gap between the SPGs and the SUPG is 1.0.
I think I understand that, but am still not sure if your statement at #6003, "Obviously the theoretical gap maximum is 1.5..." is supported by your added explanations.
Without those explanations you might think the answer is 42 . I gave a technique for simple comparisons but it doesn't add up to prove that 1.5 is the maximum. Seeing #6015 I believe I was probably too pessimistic! #6002 is well known but I never thought of it as a LUPG. Not surprising - I just invented it!
Here's a position that I created which is supposed to have a SUPG, although I feel like there could be alternate ways.
Since it's not a particularly difficult position and I don't know if I have the SPG, I'll unethically withhold the intended length 
Here's a position that I created which is supposed to have a SUPG, although I feel like there could be alternate ways. (#6022)
Since it's not a particularly difficult position and I don't know if I have the SPG, I'll unethically withhold the intended length
My conclusion is that this position does not have any unique proof games. The 11.5 move SPG is not unique, and none of the proof games in 12.0, 12.5, 13.0, or 13.5 is unique.
Would you post what you have as the SUPG for the position?
A previous post by n95 (currently #5637) contained a position with a SUPG of 3, with a longer UPG of 4.5. I'm wondering if this would count as a position with a gap of 1.5, since instead of having a SPG and a longer SUPG, both methods are UPGs.