Your Remington M1903 was one of the last ones built. The receiver and barrel both date to December, 1942. The stock is a replacement from a later M1903A3, if it has an "RA" and FJA". The finish appears correct to me, but pictures would help. The "parallel cuts" on the rear receiver bridge are the marks for a later M1903A3 rear sight. Your receiver was originally made as a M1903A3. However, when targeting problems were being ironed out, it was decided to finish the receivers as standard M1903s. These are seen on Remington M1903s from roughly the 3,290,000 range until the end of production.
"Modified" is actually a term that describes ALL Remington M1903s. It has been adopted (technically incorrectly) as a collector term to describe later Remington M1903s. The Army used Remington M1903 and M1903 (Modified) as two names for the same rifle.
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--C.S. Lewis