Infanteer, I’ll tread carefully because I may be reading you completely wrong. Like my fellow Kiwi, I’m really just trying to get my head around this.

What does “practical span of command” actually mean? Is that the max a commander should/can be dealing with or the average he happens to deal with based on subunits in contact? If the latter is the case I should think it is perhaps a meaningless statistic.

In post 30 you say:

… the practical span of command for commanders is actually quite low - 1.7 subordinates committed on average to combat. This suggests that, historically, Division commanders have put forth at most 8 companies during a majority of their actions. Employment in DS and OIF seems to validate this observation.
Does ‘committed to combat’ suggest that the remaining units are held in reserve to the point that they are shelved till required, or just that they are not in contact?

Based on a ‘typical?’ WWII triangular infantry division, 1.7 Brigadiers may suggest about half the division. 1.7 CO’s may suggest half of those brigades. 1.7 OC’s may suggest half of those battalions. By this reasoning your estimate of 8 companies would be about right.

But this seems a bit simplistic. Firstly, 9 companies in contact could be one from each battalion, in which case the division commander has all his brigades and battalions in contact. (this is working up from your number 8 instead of down from 1.7)
Also, even if the commander holds subunits in reserve and has only one in contact most of the time that contact is made, that does not mean that he is not pulling the strings on the other units. He may be manoeuvring them and/or leapfrogging units in contact or moving one through the other etc.

This may be in line with what Fuchs said in post 35:

The emphasis was on "committed" in regard to "8 companies". Storr links it also to organization and appears to dismiss whatever the other units are doing (reserve, security..) as uninteresting.
From your previous post:

The figure of 1.7 is from a Dupuy study. Also citied is an unpublished DERA study which, in looking at Division activity in WWII, showed that at no time were all nine battalions of the measured division employed at the same time.
OK, that would counter my earlier mentioned alternative. I have not read the studies.

Looking further at this data, divisions only employed a majority of their forces 1/3 of the time.
Are these not the times that matter?

The so what - if 6-10 maneuver companies (and 2-4 battalions and 1-2 brigades) are all that a division commander can realistically employ at once, than the division should be designed around sustaining 6-10 maneuver companies in combat while, at the same time, making the organization as nimble and agile in combat. 6-10 maneuver companies do not likely require an additional 14-17 companies in reserve.
Is one third all a division can employ, or the most that a division commander likes to employ at any one time for the purpose of holding a reserve? So a question here could be: is ‘one up’ at division level realistic/sensible or just not required? Is ‘one up’ at division level really one up in the same sense that it is within companies and battalions? This in terms of the size of brigades and the relative distances involved.

So iff (wink to Tukhachevskii) 6 – 10 companies are realistically the most a division can employ at once, then should the division be reduced in size or (to support I think Ken and Fuchs) should the division be cut out of the hierarchy. If you cut down the 14 -17 companies in reserve to only a few, then what’s left is perhaps a descent size brigade.

I can think of a few extreme cases where divisions were fully employed (not including the desert). UK First Airborne in Arnhem. US 101st in Bastogne (I think).

I confuse me, I’ll stop here.