I took a random sample of 400 consecutive records from the surrender (at the start of the N section up to the Noonan surname).
I only considered it a double count when the surname matched, the regiment and battery/company matched, and the rest of the name matched aside from matters of an initial missing/present.
I included as dubious those cases which could easily be transcription errors in initials only and where everything else matched.
e.g. J L Newman and J W Newman are counted as different people despite having the same company in the same regiment.
Double counts:
A F Naff, captain of the 6th Mississippi infantry
H S Nealy, private of battery H of the 1st LA Artillery.
Charles Nichols, private of Coy A of the 3rd LA Infantry
John W Nichols, private of coy E of the 46th MS infantry - appears three times, twice as John W and once as J W
(dubious) I/J Noland, private of Battery E of the 1st MS artillery.
(dubious) J P/R Nelson, private of coy A of the 20th AL Inf.
Based on this, it looks like those 400 records are for no more than 395 real people. This does not seem statistically significant, though I did skip several potential double-counts.
As of the start of the campaign, campaign strength is:
PFD: 24,100
Present: 31,133
Present and Absent: 41,227
Definitely unrecoverable casualties:
Grand Gulf 3 KIA
Port Gibson: "several hundred" prisoners (assume 300?)
Raymond 100 KIA 415 Captured
Jackson casualty breakdown unknown
Champion Hill 381 KIA 2441 captured/missing
Big Black River Bridge 1700 captured, "many others" drowned trying to cross river
Lowest estimate of losses that could not possibly be in the hospital: 5430
(highest estimate: ca. 10,000)
June 30 1863 there were 29,376 aggregate present and absent in Vicksbug, though this could include people on furlough who wouldn't actually be with the army (and thus not in Vicksburg).
Claimed surrenders: 29,495.
It looks like the count of surrenders is roughly the count of aggregate present and absent, with an expected number of double counts at about 500.
Possible future analyses to follow up on this: check the June 30 1863 report for a single brigade AP and APA and then focus specifically on the entries from the regiments making up that brigade.
All I can say is that Pemberton at the beginning of Grant's had more troops under his command than Grant.
Well, yes, if you compare Pemberton's entire department to the force Grant crossed the river with on the first night (until he moved the next few disivions over), but the reason why Pemberton's entire department couldn't descent on Grant was because of Grant's entire department and also the Dept. of the Gulf which Grant was under orders to support.
The first soldier who stepped off the boat was outnumbered over 20,000 to one, until the second one stepped off...