There is a proposition that Alperton to Mornington Crescent can be achieved in four moves. However proof has not been accepted for the existance of this number.
Valid point Artaud, initial formulas have found that we do require non-zero LV for the (move=1) station, and we will somehow need to equate these cofactors into the Fronsky calculations already mentioned by Blob. The analysis by Blob is showing that the next move in avoiding the Circle line has put a strong emphasis on 'allowance of linear movement' so I agree Euston is unwise. My own extrapolations suggest we should focus on either Leyton or Loughton thus avoiding divisibile cardinal notation.
I've had a cursory look at this and have found one flawed real-world solution and one ideal-world solution that only works in complex MC phase space. The Real solution is Alperton->Acton Town->Aldgate East->Moorgate->Mornington Crescent. The Imaginary solution is Alperton->Aldgate->Moorgate->Mornington Cresent. Unfortunately this is an odd-even move movement mapping which it is trivial to prove is irreconcilable. So you can cross that solution of your list as well. But this is only a first-pass approximation. It may be possible to use a one-move Expansion on the Imaginary at thereby unite the two. The trick, of course, is to do this without forcing a Real/Complex bifurcation.
OTOH, if you can acheive it /with/ an Re/Im bifurcation, then it's been proved that a true real-only analogue move to the bifurcated move does exist - if only it can be found. Which is the most tantalising thing about this puzzle. Theorists are almost certain it /can/ be solved rigorously, but no-one has ever been able to prove it.
Just a thought, but running with Kilydd's L~ton hypothesis and selecting Loughton over Leyton due to its greater magnetic selectivity quotient, how about stringing and ghost/live bifurcation on (move=2), possibly involving a recent ghost like Aldwych, or maybe Epping. The live side of course would have to be very lively indeed - maybe Baker Street or Bank. The dead:live dichotomy would perhaps then allow a move to a vortex station such as Sloane Square (Dollis Hill being unusable for obvious reasons). This might then lead to (move=3) being a Magenta station, though not if Baker Street had been used in (m=1) of course, it being a Magenta itself. What d'you reckon ?
This is probably a stupid question, but I am puzzled as to why this has not been solved by brute force and appropriate computing power. By analogy, think of how the four-color (or should that be for-colour?) map conjecture was eventually proved by reducing the problem to a finite (relatively small) number of maps and then checking them all. Here, at the level of actual moves, there is only a finite (and relatively small) number of four move combinations.
I'm aware, of course, that this is a simplification, because more than the pure moves come into play: there is a large (though still finite) set of possible initial placements on the board; token loadings can take on a large (though still finite) number of values; n-furcations are possible; and there are several continuous variables to worry about, such as LV and token weightings. Still, if this four-move combination is not a measure-zero possibility, one would expect that even a crude algorithm could easily limit the search space and pin down permissible ranges for LV and token weightings.
I must be missing something, but I'm not sure what.