Two ways to tell:
1) The stock has bolts, rather than pins. Early Remington M1903A3s had stocks with pins.
2) The RA and FJA: These weren't added until about mid-1943.
I would agree the rifle was overhauled at some point.
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I agree with Rick. The cross bolts replaced the pins in the Summer of 43 so it has been restocked at some point in time. I also believe it went through post war rebuild possibly more than once as you have an OG stamp and an extra P. I have never seen a an OG stock with a extra P. The rifle has many parkerized parts which originally were blued. It should have a 4 groove barrel....have you checked it? All that aside, I like the rifle and it should make a fine shooter. You can always find an original rifle just remember it should have a parkerized action and barrel and all the other parts including the complete bolt, on/off switch, bands, floorplate, sights, swivels, follower and butt plate blued.
Added: I just noticed the fine picture of your bore and it indeed a 4 Land & groove barrel that looks excellent. There are those that say there is no difference in the accuracy between a 4 or 2 groove barrel and then there is me and I firmly disagree.
They did have the FJA ... but later Remington M1903A3s, also had an "RA" stamp.
Rick I was commenting on the FJA did not start until the middle of 43.
[QUOTE=Detroit-1;433646]I thought the safety was parked on a 03A3.[COLOR="black"]
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Detroit, you are on to something and I agree, many of the safeties on a Remington 1903A3 are parkerized. I own 3 stone mint examples and they all have parkerized safeties. SN 3831092, 6-43, 4134868, 12-43 and SN4206159, 12-43. All 3 bolts and this spare have the identical safety. The first 2 pictures taken in full sunlight and the 3d picture taken with the flash on with Led light. There is no mistake, all four bolts have the same colored safety. The sub who made these parkerized them. These are all 100% original rifles never jibbered with.
https://www.milsurps.com/images/impo...06/safe1-1.jpg
https://www.milsurps.com/images/impo...06/safe2-1.jpg
https://www.milsurps.com/images/impo...06/safe3-1.jpg