Re: Application for Naturalization #general
I would send a copy of the index card to the National Archives in New York City (newyork.archives@...) and ask them for help locating the Petition for Naturalization.
FYI, the number at the top right of the naturalization index card (6986864) is the naturalization certificate number and the USCIS C-file number. Just put a "C" in front of it to make the C-file number (C-6986864). Because the naturalization (date of admission) was prior to 4/1/1956, the USCIS naturalization records will be in this C-file, which you can requesting from USCIS using the C-file number (www.uscis.gov/genealogy). There is a $65 fee.
The Alien Registration number should also relate to a 1940 Alien Registration Form, also available from USCIS for a fee. BUT there's always a chance the same form will also be inside the C-file.