While I was browsing through the VAT with CalcSys, I noticed that there were 2 'empty programs'. They were named '#' and '!', and both were deemed to be 0 bytes in size by CalcSys. I've never encountered such programs up until now, and I've never heard of such programs either. I've tried getting rid of them through a RAM clear, but they still appear in CalcSys.

Does anyone know what's going on here, or how to get rid of the programs?

Those programs contain information like the homescreen entries. Don't mess with them unless you know what you're doing, or if you don't mind getting a RAM clear.

Ah, thanks Souvik . I first thought those programs were an offshoot of a glitch in my program.

I was about to delete them in CalcSys, but I guess I shouldn't now

Correct, you shouldn't. As Souvik accurately stated, they're used to store the previous entries and the current entry on the homescreen.

Hmm... does this happen to have anything to do with the OS's interpreter? My guess is that the TI-OS runs the contents of those programs through ParseInp when evaluating math expressions on the homescreen. Or maybe I'm just getting it all wrong...

Yup, that's more or less correct, Impiety. And of course that's where it gets the previous entries when you press 2nd-Enter on the homescreen.

Wait, does the OS use an actual bcall for ParseInp, or does it do the context-switch thingy you were discussing in SAX?

Also, what the heck is a context switch, and more importantly, how is it implemented in z80?

You and the OS both do a bcall() to start executing a program using _ParseInp, which causes the OS to switch context, execute the program, then switch back. Keep in mind that a bcall() is mainly just a way to make an off-page call, nothing terribly super-special. It's less distinguished than a system call in the *nix/POSIX model.
The TI-OS has a system it calls the Monitor that keeps track of the current context: running an App, executing a program, drawing a graph, etc. It's nothing like context-switching in a multitasking scheduler on a more complex device.