Some of the PhD positions I'm applying for (all in Europe, mostly UK, Germany, and Switzerland), require a research proposal as a mandatory part of the application. I'm confused about what exactly they are asking for because the guidelines are quite vague.
I've seen this question and this one, but there are two things that are different in my case: First, the proposal for some of these programs is actually a mandatory requirement for the formal application (not for the first contact with the potential advisor), and second, unlike the former of those two questions, my field is mathematical and computational science, not Social Science (not sure if that makes a significant difference). I've also seen this question, but the question and the answers are not about how detailed the proposal should be.
My broader research interests, including a somewhat narrow area of interest for my PhD thesis (but not as narrow as a research topic), are already stated in both my statement letter (for formal application) and my contact with the potential advisor. That makes me think the proposal they expect to receive shouldn't be just my research interests, at least for those programs where, according to the guideline, it is recommended to cite related literature in my proposal. On the other hand, I really can't state a question that would be a suitable driving question for a PhD project on my own without enough help from a professor at this stage, and even if I could, I highly doubt that would be something my prospective advisor would want me to work on. So, it seems to me that writing a proper research proposal, without a substantial contribution from the advisor is impractical. (In fact, I can't even imagine how I could write about "the methodology" at this stage, considering finding the appropriate methodology in this field is often actually a very substantial part of the project.)
I'd appreciate if you could share your knowledge and experience about writing a PhD proposal as a part of the application. It would be very helpful to know what admission committees expect to see in a research proposal at the application stage, and how broad could the proposal be without turning into an SOP.