Say someone gives birth, but neither they nor their apparent partner/spouse will answer any questions about what the resulting child is supposed to be named. They are cooperative in all other ways, willing to sign off on any relevant birth certificate forms, and so on, but when asked to state or write down a name for the child, they don't.
Will the child be issued a birth certificate? If so, under what name?
Can the parents get in any kind of trouble for doing this? (Illegal failure to report a birth due to the forms not being filled out properly, child abuse due to causing obvious bureaucratic problems for the child, etc.)
The answer probably differs by state; I'm interested in answers for any particular US state.