[...] an actual minimal pair.

Example: Strauss vs. Strauß. Originally the same name, but two spellings make them two names that may need to be distinguished from each other.

True for "Wei{ß/ss}" as well. Or a non-name example: "Buße" (repentance) vs "Busse" (buses). But then, non-name examples are far less likely to remain ambiguous in context.

[...] it's not like capitalization is otherwise invertible [...]

[...] for German personal names, I would expect capitalization to be invertible, provided that “ß” has been mapped to “ẞ” U+1E9E LATIN CAPITAL LETTER SHARP S.

Yes, that would be better. Presently, some official documents retain (lowercase) "ß" within all-caps writing in some places where it really matters, but it's rare to see such a style elsewhere, and it's technically not permitted by our official orthography, fwiw.

Which amounts to a weird situation, because during the debates surrounding German orthographic reform 1-2 decades ago, one "argument" presented to those who were against was that the official body of rules ("amtliche Regelung") was binding only for schoolchildren and civil servants within government offices anyways :-) (See also this German supreme court decision <http://www.bverfg.de/pressemitteilungen/bvg06-042.html>.)

Stephan

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