I stand by position
4. Don't put it on hold because it's fine for question (B).
I don't think it's a check-my-work question, though admittedly this is a subtle point. The OP has found a textbook problem and has presumably worked out the solution, but the question is not about solving the textbook problem. More specifically, the textbook problem is of the form
Prove property P(A) for arbitrary objects A
(where A is, in that case the function F). The OP's question has little to do with the proof; instead, the OP asks whether the property even makes sense. The post is of the form
I found an object A1 such that the property P(A1) doesn't make sense. Is this known/expected? How do I deal with it?
As it turns out, the handling of that singularity is not trivial, and needs the proper framework to work well. This means a level of mathematical sophistication that is not very usual in intro-to-QM books (Galindo & Pascual being the only exception I know of), or a set of heuristics which involve not caring about specific infinities and singularities. These heuristics are particular to physics and form the core of any good answer to the question.
On a similar vein, I don't think this question is a maths question. The heuristics are particular to physics and a mathematician would either (a) throw a fit at the nonchalant use of divergent integrals, or (b) have a particularly hard time phrasing their explanation in terms that a physicist would find useful. The process in (b) is in fact that of turning mathematics into physics. Hence the question is physics.
As I stated before, calling a question "that's just pure maths" just because it has mathematical formulae in it is not productive, not least because that encompasses all of physics.
I don't find it particularly unclear, but it's not a model of clarity either. It was clear enough that it found an answer within 24 hours, so I think it definitely deserved the benefit of the doubt.
Furthermore, I claim that as regards questions that are unclear, we as a community do tend to give the benefit of the doubt to users, and particularly newcomers. I think the review queues do a good job of putting unclear questions on the plate of the people best suited to work with the OPs to clarify them and make them answerable, when that is possible.
I would of course be willing to consider data beyond my personal experience. This, however, is a separate issue.
To put things in contrast, question (A) is very different.
It too starts with a textbook(-style) problem, which reads
Given objects B, and C, what is property Q(B, C)?
and the OPs question is exclusively
what is property Q(B, C)?
This is why question (A) is a check-my-work problem. I don't find it that terrible, because of the level of sophistication of the problem and the given solution, but I still agree that (i) it is closable under the current policy and (ii) it should be closed for that reason.
On the question of whether question (A) is too close to pure math for this site, I think it is close to the fence that it's on-topic both here and at maths.SE, but its content is still relevant enough to undergraduate-level analytical mechanics that it's on topic here (as far as the physics/maths divide goes).
The main confusion in DanielSank's question is that this is even an issue. I don't think anyone, on the whole discussion, ever claimed that this question was off-topic for being too mathsy. (Of course, I'll be glad to set straight on this if someone has done so.)
To summarize, I don't think the questions under consideration show any inconsistency in our current closure practices. This doesn't mean that we are perfectly consistent, but I think perfect consistency is a little too much to ask of a system run by many individuals with varying opinions.
That said, there is probably more room in the system for more detailed comments on questions by people voting to close. This has also come up recently on a related topic. More detailed comments on why a question is getting closed helps the OP in improving their post (or understanding the scope of the site), and it improves the site in making it clear to future users why some content is on-topic and other content is not.
Our review queue is quiet enough, and we don't have the sort of problem that SO has. We have a big enough pool of reviewers that we can spare the extra time for that.
On the other hand, sorting through the queue on busy times can be a bit of a dull job, and I would ask that some consideration be shown to reviewers, as it's hard to really grok the process until you see it in action.