When does beta become alpha software?

Beta doesn't become Alpha ... it's the other way around.

First, there's the development or "pre-alpha" versions - this is when your actual development happens.

Alpha is the first working version of the software, the first thing anybody else but the developer could test. But is still in active development and trivial bugs are expected.

Beta is the first feature-complete version of the software. No new functionalities are added at this point and it's now relatively non-obvious bugs that you're after.

Release candidate is something a bit more confusing, because it may as well be the final version. RCs are supposed to be suitable for use by the general public, but at the same time being a last-resort effort to find hidden/subtle/hard-to-spot bugs, and you mostly rely on actualy users reporting them.

When you no longer receive reasonable bug reports, that last RC is "tagged" as the final version and your job is done.