Is that incorrect, or am I missing your point completely?

Sort of, but it's damn difficult explaining this stuff back and forth in text like this. I should set up an EvC Skype account

What we are doing is trying to construct a 4d path through space-time for each of our two twins, and then measuring the length of these paths. Artificially arranging a constant velocity trip to some star system, using infinite accelerations to generate that velocity, to reverse the direction at the star system, and then bring the twin to rest back at earth is one way to make life easy in working out the length of the total 4d path. But to say that the length of the path "depends" on the velocity at an particular point, is like... well, like the following:

start at a point. Move distance

. Turn through an angle

and reverse for a distance

. How far are you from the origin? Answer

So does your distance from the origin depend on

? Well, sort of, but only if that is how you arrange your path. You could have spiralled out from your starting point for a distance

, with say a widening of

per revolution. Now the relation between

and your distance back to the origin is totally different.

What I'm trying to say, is that you are getting hung up on the precise way you are arranging your particular path through space-time that you are attributing to it importance that just isn't there.

If you assume the infinite accelerations, then you are in the nice position of being able to work out the time difference from the velocities. But that is because you have given yourself these infinite accelerations to make the situation easy. The time dilation itself is not arising from the velocities even though we are using the values of the velocities to work out the total time difference.

Am I making any sense?