Context: Some years ago I collaborated on a paper, with my advisor and another grad student (first author), in which we would include
a certain competitive machine learning algorithm and
a formalization of the representation (model) that the algorithm used internally, which was a model introduced by the first author in a previous paper, with some new theoretical analysis and guarantees.
In the end, our advisor decided to split this into two papers (an idea with which I disagreed at the time since it seemed like trying to inflate our number of publications), and we got paper #1 sent and later rejected (two reasons: lack of theoretical soundness and low impact). AFAIK, they never started working on paper #2. Paper #1 has been on hold for years but all of us have agreed that "some time in the future" we will try to improve it and re-submit it. At one point, my advisor and I asked the first author if he would be willing to let me do the theoretical analysis and include it in the same paper (#1), but he didn't agree, mainly because we have conflicting ways of conceptualizing the model.
At present: I no longer work with any of the co-authors, but with my previous knowledge and some work I devised a number of proofs that may merit a paper on their own, i.e., paper #2 (not the greatest impact since it's all theoretical and very specific, but I believe it's still a meaningful contribution to the area). This paper would answer all the questions about soundness that (partially) got our first paper rejected. It might even allow us to get paper #1 published later. The problem is that, even though the ideas are all my own, I'm a bit worried about the ethical implications, since the other two authors had the explicit intention of working on the same idea (which I don't think should limit my freedom to publish my own ideas, but who knows?).
Also, I need to make some corrections in the way the ideas are described: there is at least one plain error and there is a certain idea they proposed that explains part of the models used, but this idea is confusing and does not even justify the model theoretically. Moreover, reviewers mentioned that it made no sense. I find the idea a bit "embarrassing", but the model itself is sound, and I don't want to make anyone look bad, so I just want to try to avoid that concept and simply put forward my own conceptualization and explanations. The problem is that I may come across as taking credit for the part of the formalization that was already defined in the first author's previous paper. Avoiding this is complicated, since the paper claims that the model is explained by this "strange" idea. They even wrote a misleading algorithm that is said to use this idea, but in reality the implemented algorithm works a bit differently and uses previous methods (I implemented it, and they both have full knowledge of how it works). My explanation establishes a link with previous concepts in the area and formulates an equivalence with other models. It explains how the previous methods are used by algorithms in order to build these models and why they are correct. If I try to get validation from the other authors, the first author will most likely disagree with my ideas or ask me to send him the manuscript. The other author may agree with me but will probably want to be a co-author and make things incredibly harder (based on my experience working with him). I may be wrong, but I believe that the worst case scenario is that they may try to prevent me from publishing, and the best case scenario is that they will try to be co-authors (even though they didn't come up with any of the contributions and I'm pretty sure I can make a decent paper on my own), and they may even try to change my concepts back to the "faulty" or misleading concepts. I've already had endless debates with the first author on these models, which lead to nowhere, and that's unlikely to change.
So my questions are:
Is it ethical to go forward and try to publish my paper?
How do I "objectively" and politely explain that my proofs are my own contribution, given that previous works have claimed that they explain the same thing but failed to do so or did it poorly?