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Given that a copyright application protects an author and has the "Pseudonymous" option, the legal claimant is always required in the form if you intend to have any legal protections, correct?
But a copyright is public information, like a patent. Often, inventors do not seek a patent for this very reason. Is this dilemma paralleled in copyrights? Can a pen name be both protected and not publicly outed, or are truly unknown pseudonyms in fact impossible?
You do not need to include your legal name in the registration record if you are registering an “anonymous work” or a “pseudonymous work.” . . .
A pseudonymous work is a work where “the author is identified under a fictitious name” on the copies or phonorecords of the work, such as a pen name, a stage name, or other pseudonym. Nicknames and other diminutive forms of legal names are not considered pseudonyms. Note, however, if the author’s real name appears anywhere on the copies or phonorecords of the work (including the copyright notice), the work is not pseudonymous. In this case, the applicant should provide the author’s real name in the application.
You can register a copyright under a pen name with the US government, but you have to provide a real address. If you register under a pseudonym or anonymously, it alters the term of the copyright protections, from date after death of the author to a date determined by first publication or creation of the piece.
If you are really paranoid about someone linking you with the writing, you can transfer the rights to a corporation and let them hold the copyright. It is a tax hassel, and the articles of incorporation will show your name or someone you trust as the corporate officers.
Be aware that even if you don't explicitly register the copyright, you're still presumably intending to publish the work. Depending on the size of your work corpus, and how much people care about unmasking the anonymous work, it's somewhere between hard and impossible to remain anonymous. Stylometric analysis can often narrow down the set of plausible authors of a work significantly, and from there good old fashioned investigation can often result in a single name. (See e.g. Primary Colors.) In the worst case, publishing something anonymously or pseudonymously that you wish to remain that way reduces your future ability to publish under your own name, as that would provide a corpus.