Looking at the limited pictures, I would have questions about the bolt body. It appears to be a straight handle smooth body, (no machine cuts around the safety lug). Some early Remington 1903's had a semi smooth bolt body, but not a straight handle and too early for a 03A3.
The rest of the bolt parts appear correct but with the limited info and pictures it's hard to tell. I would look at the root area on the underside of the bolt handle and see what if anything is stamped there.
The back side of the safety switch, the cut off switch, the firing pin AND the bolt root should have an R stamped on them. The trigger guard would have an R stamped on the back side of the magazine box. Sling swivels would have R stampd on the swivel flat and the rear sling swivel on the base. The bayonet lug would be stamped with an R, and possibly the front sight band.
There are original as issued, re-arsenaled, corrected, and who knows examples out there. The box and paperwork are a plus as they confirm part of the history and eliminate many of the undocumented stories that run rampent.