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Kernel means "core". Kernel-mode-only kernel is a contradiction. If you don't have separate parts, you can't have a core.

So if you write core software for an architecture without privsep you must not call it a kernel?

I'm not talking about privsep, just different modules. I gonna answer you with another question:

Did MS-DOS had a kernel?

Yes. We all lose here today.

It depends. Being downvoted by people that believes a memcpy() is a kernel, is victory.