I had an idea a while ago for a case system that includes one case
that just means "not an argument of the verb", while nouns in any
other case are assumed to be direct participants in a verbs argument
structure. It would be used for adverbial adjuncts, genitives, and
objects of adpositions. In such a system one might have, for example,
a locative case which could only be used with verbs that select for a
locative argument of some sort. Expressing location as an attribute of
some other noun would require some other construction (possibly an
adposition with "not-an-argument" case).

I got thinking about this again last night and decided that it seemed
a little odd (not at all unnatural, just odd) because there's this one
case that doesn't behave quite like all the others; perhaps it
shouldn't really be a case at all. Perhaps it makes more sense for
"not-an-argument" to be expressed by a difference in state, orthogonal
to case.

That would allow "not-an-argument" things to be marked with other
cases, separately from their status as arguments, which I think turns
out to actually have some neat disambiguatory uses. Most obviously, it
allows you to distinguish adverbial and adjectival adpositional

 "I saw the man with a telescope." -> "I used a telescope and saw the man."
 "I saw the man with a telescope-NAA." -> "I saw the man who had a telescope."