I'll go with Ballard as the Vicksburg expert.
Okay, so does Ballard cite his numbers? It looks like he cites a quote.
The fact remains that we have OR reports of the strength of the units that crossed, and as you'll see in the quote Logan crossed in "the early morning hours of May 1" so they're part of the same body.
Perhaps Ballard is counting Grant's force in "effectives", but if so then we should reduce down Pemberton's troop strength in the same way (which looks like a reduction to 71% of PFD)
The crossing was certainly an accomplishment - moving 30,000 troops PFD in one go is pretty impressive - but I cannot see the metric where one gets from the ORs, looks at the troops who moved, and concludes that the force Grant crossed with was smaller than the force Pemberton had within fifty miles.
How does this show the focus of Union strategy? How many troops per square mile of the areas these armies covered? How many troops per army as compared to their confederate counterparts?
On April 30, 1863 (which isn't quite comparable to the Union numbers, I admit) the strengths Present are:
AoNV (plus Hood and Pickett's divisions) 94,000 - this excludes Ransom, Imboden and Davidson
Army of Tennessee 68,000
Dept. of Mississippi and East Louisiana 61,000 (this isn't a field army but it faces Grant's one)
Dept. of South Carolina, Georgia and Florida 37,000
Trans-Mississippi dept. 30,000
Dept. of North Carolina (which includes Ransom's division) 27,000
What this means is that the largest Confederate field army is in the East and the largest Union field army is in the West. There's one big Confederate department after the three largest which is in the West, and two big Confederate departments after the three largest in the East.
Total Confederate strength in the West (cis-Mississippi) is 152,000 counting Maury's little Army of the Tennessee and the Dept. of the Gulf, as against 235,000 in the two big Union field armies operating there (Tennessee and Cumberland).
The Confederate main army in the East is at least 100,000 if Ransom's division is counted, and the Union field army there is 118,000 Present.
That means that the Army of Northern Virginia alone is bigger compared to the Army of the Potomac than just about everything in the West compared to the Union field armies there.