I think I've probably come across some of your Antietam posts before, these numbers look pretty familiar to me. It always seemed really strange to me that Lee supposedly had ~38,000 men at Antietam, but in virtually every other battle he tended to have 50-60,000+ until you get to the Appomattox campaign. Why would he go looking for a fight on Northern soil with an army that relatively small, I always wondered.
As far as we can tell the answer is straggling, plus definitions, plus some cases of dodgy counting.
The straggling one is easy enough to explain - the number of men actually in Lee's moving army in total was about 75,000 (PFD) which is reached from three independent metrics, but large numbers of men were sick or dropped out temporarily on a high speed march. Anyone who dropped out on the march from South Mountain to Antietam would have just been captured, but anyone who was in the wing that marched to Antietam south of the Potomac and who dropped out on the high speed march wouldn't have been. Antietam saw both armies straggle heavily, and the problem is that McClellan gave his strength
before straggling (because it was the number he had) while Lee didn't have any such number to give.
The second problem is definitions.
Regulation methods of reporting strength at the time at a high level were PFD, AP and AP&A.
PFD is all men present for duty (reported separately for officers and men) where "present for duty" refers to those able to fight in the line of battle plus those with certain other duties.
AP is Aggregate Present and refers to all men actually
with the army; this includes those on more types of duty than PFD, and (IIRC) it also includes men who are sick. This can also be thought of as "ration strength" or "pay strength".
And AP&A is Aggregate Present and Absent, which refers to Aggregate Present plus the total of men who are absent from the army with or without leave. Late in the war this distinction became huge - as of early 1865 a third of the entire Union army was absent with or without leave.
The Confederates (at least in the East) did something slightly different, possibly because a lot of their logistics duties were covered by (put delicately) black men, many of whom did not want to be there. They developed a tendency to report "effectives", and what they meant by this varies but it can be generally thought of as "the number of men actually able to fight in the line of battle"; it's less than PFD and can be much less.
Over time the extra-duty men in the Union army were shifted out of the PFD column, making the two armies more comparable in their strength measures, but this hadn't happened as of Antietam.
The third problem is that the Confederates often used somewhat dodgy counting methods and these are often the most widely reported values we have for Antietam. For example, Jackson's division is listed according to a certain strength it had pretty much exactly when it arrived on the battlefield; however, this was most of a day before they fought and there is positive evidence that many of the men who had fallen out on the forced-march came back up again and made up the numbers. (Jackson's division is given as 1,784 infantry and 310 artillery, which is the state on the afternoon of the 16th; the acting commander of Starkes' brigade said that his brigade alone had 1,400 to 1,500 men in action, and Starkes' brigade may have been the largest of Jackson's four brigades but not by
that much...)
This played into a natural tendency for the CSA to try and emphasize how much they'd only been beaten by overwhelming force clumsily applied by idiot Union generals, which was to take a large measure of Union strength and match it against a small Confederate one so as to imply that Confederates were superhuman compared to Union troops.
So long as you're comparing numbers obtained by similar derivations for both sides, it's fine; the problem comes when you count troops by one method for one side and the other method for the other side. So Lee's force in the Maryland campaign was about 75,000 PFD before straggling, and McClellan's was about 87,000 PFD before straggling; after straggling and/or reducing to effectives the picture was probably about the same, with McClellan's force a little larger and stragglers littering the approach roads.
If nothing else, this does explain why exactly Lee thought it was worthwhile invading the North, and why Lee's October 10 returns give his strength somewhat north of 60,000 - which would have made invading when he did
incredibly stupid if he'd really only taken 40,000 men across the Potomac. Why take 40,000 men north when you could wait and recieve an extra 35,000 reinforcements?
Interestingly, this point about different definitions also speaks to what casualties mean.
Short term, an army with 40,000 effectives which loses 15,000 casualties has suffered a body blow; long term, an army with 75,000 PFD which loses 15,000 casualties can recover quite substantially because it can shift men out of the logistics that supported the now-missing troops (which would show up on regimental returns as a reduction in the number of extra-duty men and an increase in the line strength), though of course caring for the wounded can also soak up a large number of men which is one reason why hospitals were so important.