Hi Whig,
You have a very nice Smith Corona that has a mix of Remington parts thrown in.
The Stock would appear to be an early Smith Corona stock by the use of stock re-enforcing pins instead of screws/bolts. Additionally, your Smith Corona band spring inlets are rounded and correct instead of being squared as seen on Remington stocks.
According to Joe Poyer's book about the 1903...Smith Corona rear sights were generally unmarked.
Joe's book also points out that the RA-P mark indicates Post-World War II rebuild at the Raritan Arsenal.
Attached is a Smithy, like yours, that sold for $2,875 at an RIA auction back in 12/3/2016!
(You need to Login to view media files and links)Also attached are Pics of my Remington (top) and Smith Corona (bottom) Model 03-A3s. Both of them came to me originally sporterized.
I estimate that my Model 03-A3s would sell for around $600-800. Yours is an original with a scarce 6 grove barrel and would command a much higher price!