Longshanks

You may not, as you say, be the master of the battlefield, but you are the master of organization. Without threads like this we would have no idea where all the information is

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Jim

I followed your links and the links in those links and the links in the links in those links and ... I learned a lot. There's a lot of goings-on from when AACW was still having a lot of dents hammered out and from the heroes of day-one.
Pat

Knew every answer, save one. This is why Pat has earned the greatest respect as an opponent and as a contributor to this forum, plus his avatar is really handsom

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Lodilefty

You always seem to be there to fill in the blanks, put things into perspective, set things on the right path ... is there anything you don't do to make this game and this forum all that much greater? I wonder some times where you find the time and energy.
Were this not an historical game I would suggest that your names be honorarily be given to generals and... oh wait, Pat already has one

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Okay, enough of the laudatory.
To answer my own questions, MTSG is an abstract of the Real-World™ military action of one force coming to the aid of another. The game calculates whether the marching force would arrive on time to partake in a battle, taking stance, distance, obstacles, leadership traits and everything affecting movement into account to calculate the percentage of success, and makes a percentage Die Role to determine success or failure.
If the DR is successful, that force takes part in the battle.
If not, it doesn't.
Once the battle is over, there is a small chance that the marching force cannot return on time to its original location (depending on the distance and length of the battle and when the battle actually took place from my experience), in which case it remains in the battle region.
Because this concept is an abstract, the actual movement does not take place in game turns other than the marching force 'being' in the battle location, a force can MTSG to different battles in the same turn, which actually take place on or near the same day. It is abstract.
This is the reason that Jackson's Corp didn't seem to do more than just pause for the day of the battle in Corinth, because that's what it actually did.
Why he didn't partake in the Memphis battle I will have to assume lays with fate. Jackson having fought in Corinth would have lost some cohesion and probably having finally reached Tipton, in game terms, lost some there too. These would contribute to the DR to determine Jackson's ability to MTSG again. As stated, unless the percentage chance of a force to MTSG is 100%, there is still a chance that it won't. My luck in that case.
Ergo WAD.