Mansion House to Mornington Crescent is possible in exactly five moves if LV>6.4 and Bank Holiday rules apply. I have a really simple proof for this, but my heart isn't what it waaaaaa....
Right, so starting at Mansion House, and getting to MC in 5. Let's see... 1: Moorgate -- Ah, that's closed the diagonals. So we can't start this way... maybe someone else wants to give it a try.
Can the first move be 1 I would suggest that we strat at 0, however, seeing as you speculated as to the first integar. There is doubt as to whether it now can follow that n=2. Pimlico invoking Flach.
I would personally look at 1. Kew Gardens, which tells me we need to solve some pretty tough integrals to get any further, but which aren't impossible. String theory might also prove helpful.