In this context, maybe a phrase, such as well-studied would be good. It implies that the mathematical objects have been known for a while, but does not have any negative connotations (to me), nor implies great antiquity. I think it depends a lot on the context. Had you asked for a similar word to describe cheese, I'd have suggested aged or well aged, but I don't think that applies at all in this case. ;)
I re-wrote one of your sentences to not use the word "old". In the other two contexts, I honestly don't see anything wrong with it.
The Foo sequence is named after John Foo, but it is actually the same sequence as earlier
discovered by Amy Bar in 1945.
[Note: I probably should have gone more into differences in specific meanings in context, but I feel like that would be too time-consuming. If necessary, I can add something in later.]
I started thinking of what might replace the verb phrase in some examples that came to mind, with minimal rewording of everything else.
Notice that "fall short of" refers to not reaching some sort of fixed standard. Consequently, the OED suggestion of trail doesn't really work that well, because something trails people or things over a course of time -- for instance company X's stock trails that of company Y, whereas there is sometimes a single-shot element to "fall short of".
Let's look at the examples again and things that could replace "fall short of".
- All fall short of the glory of God.
- She falls short of passing her matriculation exams.
Something like "cannot reach" (with the appropriate inflection) would work with (1). But it does not for (2), since in that case, the appropriate wording would be something like "she cannot reach the passing mark on her matriculation exams"; we cannot simply replace "falls short of" with "cannot reach" there.
Meanwhile, "fail" could work as a replacement in (2), but we would also have to reword "passing" to "to pass", since the gerund (at least in this sort of construction) requires some sort of word between it and the inflected verb. Your mileage may vary as to whether needing additional rewording is acceptable behaviour or not.
But this doesn't work in the OED example of:
The other Prophets fell so much short of Moses.
The other prophets don't actually "fail" Moses -- something which would suggest that they had some sort of obligation to him that they do not actually meet. Rather, they simply fail to reach the standard set by Moses. Again, without a lot more rewording beyond simply changing "fall short of" to something else (via expanding on "Moses"), there again, like in (2) doesn't seem to be an obvious one-word replacement here.
As alluded to by FumbleFingers, this seems to suggest that the correct synonym is context-dependent, because of how widely "fall short of" can be used. This isn't just because of the grammatical variation in what might follow "fall short of", but also is an issue of what sort of thing one is actually "falling short of".
In terms of meaning, "fail" seems to work, except we can't simply replace "fall short of" with it and expect everything to be fine, even once superficial issues of grammaticality are dealt with (because meanings can change as a result of this).
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