Your stock is made for a Remington 1903. Not the 1903A3, although the 1903 will fit in the A3 stock, however an A3 will not fit into an 03 stock. The left side of your stock has overlapping stamped markings that kind of blend together. Hopefully the pictures can clear up any confusion.
Attachment 91480
The RA-P as already mentioned is a rebuild stamp for the Raritan Arsenal. Probably went through there toward the end of the war.
Attachment 91481
Underneath the RA-P is the RA(Remington) and the boxed FJA is the Ordnance inspector's initials (Frank J. Attwood).
Attachment 91488
What makes it a 1903 stock instead of a A3 stock is the lack of the handguard ring cutout just forward of the receiver on the stock.
BEAR