Fatigue, hardship, stress, etc. The numbers do not qualify as a given specific.
No, they don't, at least not explicitly. But the positive value of CEV is that, for example, it can let us put two battles side by side - or two careers. And, most importantly, it explicitly includes troop strength, which I think almost anyone would agree is a critical metric for determining how good a commander is given the result of a battle.
Clausewitz himself wrote about how winning when you have a 2:1 superiority (in broadly similar troops) is very easy indeed and winning when you have a 2:1
inferiority is a good performance and/or shows that your opponent was rather poor; naturally this is modified by defensive positions and so on.
CEVs are useful because they allow us to get a baseline for comparison - that's all.
Going beyond that is where one starts to bring in nuance. For example, in analyzing Antietam, it's possible to say that:
CEV-wise, it was the most successful attack launched on the Army of Northern Virginia.
This could be attributed to the way that the AoNV had not fully dug in, or it could be attributed to the hard marching fatiguing the Confederate troops, or it could be attributed to any one of a number of other things. But what the CEV illustrates is that one cannot sensibly say that Antietam was really a wasted opportunity - the Union did more damage to the AoNV relative to the amount it sustained than in any other battle in the war save perhaps for Gettysburg, and what's more the fighting at Antietam was the most compressed single day of violence in the war. So the Union army attacked Lee, and did him a lot of damage; if this isn't sufficient then nor is almost any battle in the Overland just for starters.
Similarly, one can look at Fredericksburg and say what we already knew - it was a very bloody battle and the Union took a lot of casualties. They were unable to effectively leverage their newly larger army.
As to why the CEV was so poor? The fact the Confederates were dug in is some of it, but beyond that you have to look at, say, the idea that it was because the assaults on Marye's Heights were made by brigade (and thus allowed all the Confederate defenders to engage each individual attacking wave)
The CEV is the beginning of analysis, not the end, but it does no good to dismiss it entirely.
Sure is good as a litmus test of some sort to properly set it side by side, but an army of confederates scattered like chaff in the wind can qualify as an aggregate number for determining strength....Is there a number for NO?
I'm not sure what you mean by "NO".
As for how the CEV calculation incorporates a
routed army, it can be calculated including the troops who don't make it to safety as casualties for the victorious side. For example Appomattox can be calculated as a CEV of about 1.25 (eyeballed) if you do it that way, which is Grant's best in the East by far and is good but not "amazing" (since Grant outnumbered Lee about 2:1 and so forcing the surrender of the enemy army at a cost of 10% of your own force is "better than expected but not stupendous"; if Grant had compelled the enemy to surrender by outmanoeuvering them without fighting, the CEV would be sky-high.)
It's also possible, though a little trickier, to look at the CEVs for individual engagements. It would be theoretically possible to compare the strength (on the field) of each Union corps at Antietam with the similar strength of the units that engaged them, and thus calculate which of the corps assaults (1st, 12th, 2nd, 9th) was handled best; the main thing that would make this fiddly is if there were Confederate brigades involved in multiple assaults.