This is not the case. If registration were required in the US, 99.999999% of software ever written would be effectively public domain.
Now, fixing a creative work in a tangible medium is all that's required. When does the copyright expire? Nobody will know, because there's no year of publication listed, and no author listed to find out when they die. (Even if there is an author listed by name, maybe it was me; maybe it was the Pulitzer Prize winning author)
This has not been true for a very long time.
Software can simultaneously be covered by copyright, trade secret and patents. The patents have to disclose some info, of course.
Even when distributed you can distribute just the binary, and keep the source a trade secret.