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Following a thread on german.SE I wondered why ει μη is translated as German "außer" (other than, except; translated as "unless" in one of the links) Epistle to the Romans (13, 1).

The wiktionary tells me for the individual elements that "μή is the negative of thought or wish" and that ει is a kind of relative conjunction. That is, it would be relative-pronoun-like in spirit of the etymology. This synthesyzing is probably my main problem, but I am also troubled because there are no direct cognates in Germanic, as far as I am aware. The naive dictionary translation "if not" matches the translation at hand, but I wonder how biased this is by attempts to fit non-Greek grammars. The element of "thought or wish" at least is missing completely. It doesn't help that etymologies of particles are often so convoluted as it seems.

BTW: ei me apo is then translated as except of (Ger "von"), but apo well corresponds to off (Ger ab often combined with von, think get off of that horse). Whether "ohne von" was a set phrase or a new coinage by Luther, now modernized (or corrected) to außer von is obviously not in question here, but a notable tangent.

I couldn't help but notice that ei me somewhat rhymes with eime "to be". Since the etymology of me is uncertain, could the verse be a formulaic archaism? Could it have become corrupted?

The wiktionary tells me for the individual elements that "μή is the negative of thought or wish"

Pretty much. It's one of the two common negative elements in Ancient Greek; οὐ(κ) is generally used with indicative verbs, while μή is used with subjunctives and optatives. Μή is also extremely common in conditionals, which will imminently be relevant.

and that ει is a kind of relative conjunction

Technically true, but a bit misleading. The more relevant translation for your purposes is simply "if".

In other words, εἰ μή means literally "if not" or "unless". This combination can also be translated as "except" without really changing the meaning:

οὐ γὰρ ἔστιν ἐξουσία εἰ μὴ ὑπὸ θεοῦ

For no power exists unless [it comes] from God

For no power exists except [that which comes] from God

P.S. Following your tangent a bit, for the following word, some manuscripts say ὑπὸ, others say ἀπὸ. In this context, I like ὑπὸ more; both words fundamentally mean "from", but ὑπὸ can also mean "under the authority of", which seems right here. (On the other hand, ἀπὸ can mean "originating from", specifying the fundamental source, which also makes sense.)

P.P.S. Be careful with your transcriptions! "Unless" is εἰ μή, with an eta, while "I am" is εἰμί, with a iota. The two letters are pronounced the same in Modern Greek, but were very different in ancient times, as we can see from transcriptions into other languages.