Hello, my name is ***** ***** I will be the immigration attorney helping you today.
Unfortunately as private attorneys, we do not have records of when a person received their naturalization certificate or became naturalized. That is all private information.
The FOIA is the only way to do it for free that exists but it may takes at least 2-3 months. Since you are asking specifically for a copy of the naturalization it may not take so long.
Something I can tell you is that I have helped many clients with their Citizenship applications N-400, and there have many times where their spouse does not know their exact day of naturalization. As long as you know more or less like the month and the year, that is sufficient. The application for your wife will not be delayed or denied at all because of this so if that is the only reason she is not submitting it, it should not be. She can also tell the officer at her interview that she did not know the exact day because you misplaced the certificate. Again, I have never seen this be an issue. It is up to you but this is my professional experience.
Good luck to you during this process and I hope this answer has helped you!
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