I think Kilydd is correct in his estimation of ~tonness; it may help that looking at my Fronsky I perceive that the Central line must be avoided for at least the first two plays. I must say it was remarkably optimistic of Stanley-Smythe to even attempt initiation using the at-the-time super-foetal Beckton - the DLR being in the mid C19 a concept "beyond the imaginings of mere mortals" as Conan Doyle once referred to future line play.
By my calculations the only TON stations (excluding ghosts) on the network are :

1. ALPERTON
2. BECKTON
3. BRIXTON
4. EASTACTON*
5. EUSTON
6. HIGHBURY & ISLINGTON
7. HIGH STREET KENSINGTON
8. KENNINGTON
9. KENTON
10. LEYTON*
11. LOUGHTON*
12. NORTH ACTON*
13. PADDINGTON
14. SOUTH KENSINGTON
15. SOUTH KENTON
16. WEST ACTON*
17. WEST BROMPTON
18. WEST KENSINGTON

Then of course there is MORNINGTON CRESCENT itself - but that is hardly going to be the starting station.

Finally there are two stations on the North London Line namely

• HOMERTON
• SOUTH ACTON

But it would be a strange ruleset indeed that allowed initiation from the N.L.L.
The stations marked * above are on the Central Line and are therefore excluded by my Fronsky calculations.
I hope this little exercise will be of help.

Oh, and considering that the (move=0) station is Alperton then the (move=1) station is not likely to be a doubling move either. If Beckton has been ruled out by Stanley-Smythe as reported, then that only leaves eleven possible stations by my reckoning - and Euston would seem highly improbable too. What d'you reckon folks ?

Presumably this assumes a theoretically stable level of play (LV and snoods stable, escalators and lifts dormant, etc.)?

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.

Let me power up my Sinclair ZX81...

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 ?

Sorry, sorry, sorry - for Epping in the above, read Ongar obviously - the latter being a ghost, the former not.

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.

[CdM] Speaking only for myself; I'm too lazy to program it.

Anyway, did some more work on this today, and spotted an interesting complementary move to my Friday's suggestion.

Alperton->Kensal Green->St John's Wood->Mornington Cresent. This is just a straight horizontal. But curiously in the complex plane the sequence has five components