Is ないで a te-form?

Question

In this answer, Darius Jahandarie says that ないで is a te-form. I asked about this on chat, and Flaw answered with a question:

If auxiliary ください follows after てform

and ないでください exists, can we reversely conclude ないで is て form?

This makes sense to me, so I think it is a te-form. But a te-form of what? It doesn't look like the te-form of ない, which I think is /nak-u-te/.

Etymology would be helpful here, but I found a comment by Matt saying the etymology of ないで is basically unknown, and another comment by Tsuyoshi Ito pointing to the entry in 大辞林, which seems to agree that there is no established theory for the origin of ないで.

So perhaps the best thing to do is to focus on how the word functions and ignore etymology. Two possibilities come to mind:

1. Perhaps ないで is a word with only one form, and that form happens to be a te-form.

2. Perhaps ないで is considered an alternate te-form for ない, even if that's not what it was etymologically.

Do either of these explanations make sense? Is there a better way to explain ないで?

Answer 1

There are other usages which hint it's a て form, like the request 「食べないで」 (although this is probably just an omission of ください), 食べないでほしい, and many other 補助動詞{ほじょどうし} which you can use with it.

So, given it's a て-form, the question is why is it ないで. That で is almost certainly not the 連用形 of だ, because having a copula there does not make sense. Looking through The uses & etymology of で answer, it seems like に + -て is a major contender, but I in fact think that's not what's going on here, because the に case particle does not fit here.

My answer is that this really just is the normal 「て」助詞, but just voiced due to its proximity to the voiced い of ない.

Of course, the real question is why ないて⇒ないで is even attempted to be used rather than なくて. I think this is because it is the 助動詞「ない」 rather than the 形容詞「無い」. In fact, 食べなくてください seems wrong/weird to me. On the other hand, 食べなくてもいい requires the なくて version.

It's not terribly clear to me why some usages of negative verbs require ないで for the て-form, and others なくて, but it does seem reasonable to me to call ないで a て-form.