Peer-review in ‘fully peer-review journals’
I think most of this question should be closed as a duplicate, and I would strongly recommend the OP read all of the highly upvoted answers here before commenting further:

I believe I have solved a famous open problem. How do I convince people in the field that I am not a crank?

However perhaps the last part of the question can still be answered:


  How can the author get an honest, unbiased assessment of his work in this situation?


Have your work reviewed by a close colleague who knows you better than the strangers your work goes to through peer review. Ideally, this is what an academic advisor is for, but other people can act as surrogates in some situations. The person should be someone close to you, who is willing to spend some time reviewing your work because of that relationship, but who has the academic abilities, rigor, and qualifications to review your work.

Impress upon this person that you fear your work has some error, because otherwise it is a very important finding, and ask for their assistance in finding the error. Make it clear that you are not looking for praise or affirmation, but want to find the error. Don't tell them how long you have worked on it or say anything else that makes it seem like you have some personal need to feel good about this work: you only want to find the error.

If together you actually find no errors, and given you have followed my advice about finding someone qualified, then, given the importance of your finding, I would expect this person to make every effort to help you with explaining the work to others.