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Does anyone know the difference...

PCGS has two certification numbers for the 1861-O half dollar, for what's known as the 'CSA Obverse'. This is the strike that has the nose-to-rim die break. One of their certification numbers is '39968' and the corresponding description simply says "CSA Obverse". The other certification number is '390906', and has a fuller description on the label, which says: "CSA Obverse FS-401 (FS-007) WB-102". But practically speaking, as far as I can tell, they are the same coin! I know some have a small tine next to the upper left of the first '1' in the date, but surely that wouldn't matter--would it?
I have sent an e-mail inquiry to PCGS months ago asking about this, and they never responded.
I have sent an e-mail inquiry to PCGS months ago asking about this, and they never responded.
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For many years the Obverse w/die crack was considered the CSA variety. Research over the years has identified many more diagnostics that pin down coins that were also struck by the CSA.
I'm not speaking for any grading service but this is what I believe to be a possible answer to your question w/o me needing to spend time looking at PCGS #'s.
All the major TPGS assign catalogue numbers for the coins they slab. As new coins are added to the catalogue, new numbers are assigned. Let's take an example with 1861-O coins. When PGCS was started they had 1861-O and 1861-O CSA in the catalogue. As collectors paid for attribution, they needed to put Breen, FS, or WB #'s on the two different 1861-O coins that already had an assigned number. As new CSA varieties were discovered, they were given new numbers. So in theory, an 1861-O half struck by the CSA could have at least five different printed labels - all with a different catalogue number.
As for the different cert. numbers being used for the CSA Obverse, I could understand it if a new 'replacement number' evolved, as you seem to suggest. However, they are still using both cert. numbers to this day, so I have to wonder if PCGS has simply dropped the ball on this one.