Unique Proof Game
A composition where the solver is invited to find the complete sequence of moves from the start of the game leading to the diagram. The solution is unique subject to the stipulation, which will generally be one of three types:
(1) Proof game in x.y,
(2) Proof game in exactly x.y
(3) Shortest Proof Game (with no number of moves specified),
Strictly, "PG in x.y" is taken to assert that no shorter solution exists (unique or not). The parallel with short solutions in orthodox problems is not apt. A short solution to a unique PG can only exist for tempo reasons, which would usually increase rather than decrease the worth of the composition.
Where a shorter route to the diagram is known to exist, the stipulation "PG in exactly x.y" is used. However, it's not clear whether all the PGs marked C+ in PDB, have been checked for additional solutions in 0.5, 1.0 or 1.5 moves fewer.
The stark stipulation "SPG" is not used commonly for unique proof games these days. The sense is that by indicating the number of moves one gives a fair clue to the solver and gives a hint of the level of difficulty. It does appear with twinning compositions, or where something special is going on.
The stipulation "SPG in x.y" would make logical sense to indicate that there are genuinely no shorter solutions, but this is not current practice - instead the term "PG in x.y" is used for this.
We should note that "SPGs" and "PGs" also flourish as apparently synonymous terms for the whole sub-genre!
If there are multiple solutions or variants, then this can be shown by adding a numerical parameter: 2, 3, etc.