Should/Can solution-verification be posted as self-answers?

Question

In this answer a user suggests that "answers should almost always be posted as answers”, which I quote

If the question exists elsewhere on the site, and you have an answer for it, it should go on that question. if the question doesn't exist, you should write it up and self-answer it.

This is the best fit for our format.

The primary goal is to get feedback about your solution. And the way we get feedback for solutions is to post them as a solution, and then people have at them.

(FYI: the tag proof-verification is now identified to solution-verification, so I am using modern day terms.) This suggestion gives arises to two questions. A stronger one is:

Should solution verification (always) be posted as self-answers?

And a weaker one is:

Can solution verification be posted as be posted as self-answers?

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There are a few scenarios that I can imagine, in which I would like to ask a solution-verification question:

1. I finished the proof, and a something in the middle is contradictory to a result that I found in the other book.
2. I finished the proof, but I am not confident that the conditions to plug in a theorem are indeed justified.
3. I finished the proof, and realized that my methods also applies to a significant generalization of the original problem. Sub-cases includes:
   1. I was going to do a college-level exercise, but indeed proved a famous unresolved problem.
   2. I was going to prove $p\wedge q \to r$, but indeed proved $p\to r$. That is, some seemingly critical conditions are not used.
4. I originally had a problem with my proof, and the problem resolved as I typed in that question. But I am still only 85% sure that the whole thing is correct.

Because of the premise of (1), (2) and (3.1), I think the answer to the stronger question is “no”. But this still leaves the weaker question open.

On the one hand, it looks fruitful to post answers as answers. Because if the answer to “Is my proof stated as … correct?” is likely “Yes”, such questions are unlikely to receive a well-written answer (specifically, a well-written answer in MSE. A related opinion. How to answer proof verification questions?).

On the other hand, I haven’t seen anyone doing that. In fact, I haven’t seen many self-answers. Occasionally people write up their solutions after being hinted and/or encouraged in the comments, but the actions of simultaneously posting questions and answers are, in my perspective, rare. Moreover, it seems difficult to do that. It appears that the community has a high-quality standard for self-answers (source: e.g. this). Also, providing context can be hard. (Is self answer considered as adding sufficient context?)

I would like to know under what circumstances posting solution-verification as self-answers is allowed/encouraged, and if there have been successful examples of people doing so.

Answer 1

It's weird that you don't try to address the points made in my answer on that linked thread, which explain why all your scenarios fall under the category of solution-attempts that should not be posted as 'answers'. To emphasize again the main points:

1. This is not a forum, but a Q&A site, so the notion of "answer" should be reserved for those who are confident of their mathematical contribution to the question at hand. I've seen many many Math SE 'answers' posted by users who think they have a proof but actually don't, and many of them lie around unchallenged. People who have a very weak foundation in mathematics should not be dumping their so-called 'proof attempt' of any mathematical problem onto an existing thread asking about that problem! This is the reason for having the solution-verification tag, with the goal of distinguishing clearly between actual answers and attempts.

2. Proof-verification questions exist to serve the needs of students who need guidance in writing proofs, perhaps because they were not taught basic first-order logic, and thus cannot quite put together a logical argument from their ideas. This definitely covers all your scenarios.

3. Proof-verification questions can be closed as duplicate if roughly the same approach had been asked about before. It is the responsibility of the asker to do sufficient research to determine whether his/her approach has been tried before, and post a question only if he/she cannot find it (at least on Math SE). This probably does not apply to you.

4. A proof-verification question will not just be a request to check some proof. Rather, it would specifically identify something that distinguishes it from other known proofs (including those the asker have found on Math SE). Mistakes are also more easily identified; if the asker takes effort to highlight how his/her proof differs from the known ones, experts can much more quickly tell whether the difference is inessential or not. This criterion greatly enhances the value of the question.

Concerning the issue of answering a solution-verification question, I don't see why quid's answer is problematic to you, especially in light of the above criterion for the question to be a good one. In fact, requiring the asker to pinpoint what he/she finds most troubling often causes the asker to resolve their own question without even asking, and if they still need to ask it will indicate some significant conceptual misunderstanding that will certainly not be answered by a trivial "yes" or "no". So this approach solves that problem too.