The Baldwin class number system is the one that will identify a locomotive as the number was stamped on a number of locations such as frame, cylinder piston guide rods, rods, tender shells, cabs. The often reported construction number on the circular smokebox plate could be lost during a rebuild.
I am told that the Smithsonian has the ledger books that list both the class number and the construction number.
I am also told that the Baldwin class system started with an early identification system and then had to modified to allow for different lead and trailing trucks, etc.
Brian Norden