Move count cube states

[ThatGuy]

This is purely for curiosity. Let's take two states of the cube, 1 and 2. Lets say it takes "x" amount of moves to get from 1 to 2, using the quarter turn metric. Now, between 1 and 2 does there exist another solution such that the solution is also x moves? Can someone give a semi rigorous proof? It seems that there is only one x move solution between two states. Just starting from one move, let's say R'. There is only one move that can bridge the two states. This stuff fascinates me for some reason. Like UFT.

 

[Kirjava]

D2 B2 U2 F2 U2 B2
F2 D2 F2 D2 F2 D2

 

[ThatGuy]

D2 B2 U2 F2 U2 B2
F2 D2 F2 D2 F2 D2

oops. Time to salvage. Then what about under a certain number of QTM. IE what is the max QTM such that two bridges to a state can't be the same.

 

[Forte]

F R U' R' U' R U R' F'
U' B' R B R' U' R' U R

F R' F' R2 U'
U' R2 B' R' B
is the shortest I can find

 

[Cride5]

D2 B2 U2 F2 U2 B2
F2 D2 F2 D2 F2 D2

oops. Time to salvage. Then what about under a certain number of QTM. IE what is the max QTM such that two bridges to a state can't be the same.

If you count them as distinct moves:
R and l x (1q)

Otherwise:
R L and L R (2q)

 

[Stefan]

a. FU'R'FRF'
b. U'FRU'R'U
c. UL'U'LFU'