Co-authors either dismiss my ideas or don't understand them... and then dismiss them anyway

(1) This question could only be answered by reading your drafts and discussion carefully.

(2) If they invite you, politely decline. (If there's more going on here, for example, a debt of gratitude, complex relationships with a third party, funding ramifications, etc., please explain.)

Now I'll pose and answer the question I thought you were leading up to.

Question:

How can I deal with their stubborn refusal to read my comments carefully and give them fair consideration?

Answer:

When you are the low man on the totem pole, it is all too easy to fall into an unproductive communication game with such people. The key is to use an authoritative but not disrespectful tone, write in a neutral tone with no emotions showing, repeat yourself as needed, and time your repetitions carefully. Note about the timing: don't reply right away. You may draft an email response right away, but send it after an interval of time has gone by.

There are two ways to repeat yourself. You can choose whichever one works better for you.

(a) The succinct approach: "I responded to these points in draft number.letter, submitted on date." Or: "Please refer to comments submitted by email on date, time. Notice "comments"(not "my comments"); "submitted" (not "which I submitted"). Depersonalizing the language you use with them helps your messages sound authoritative.)

(b) Copy and paste from the previous draft or email message.

There must be no hint of frustration or sarcasm in your messages. You can be a broken record, as long as you don't show you're aware that you might sound like a broken record. And you must not show you are annoyed, because with people like this, showing annoyance is paradoxically counter-productive.

It might be easier to adopt this tone if you pretend you are the right-hand man of an expert in the field, drafting a response for the Big Researcher. (This is analogous to being a legal clerk to a big name judge.) Write your comments and email messages as though they will be signed by that imaginary senior researcher (which might be you twenty years from now!).

These tips bring no guarantee of success. But at least you'll have minimized the time and energy you've put in.

I'll try to extend the answer given by aparente001 by a small but important point: Give constructive critic.

Taking your example, if possible don't say "I think 2+2=5 is false, we should reconsider," but rather take the reconsidering into your own hands: "I have appended a correction of claim X and proven that 2+2=4. By changing our paper slightly in the ways I list below, we can still keep the main result intact."

By doing it like this, you don't force your co-authors to think about why 2+2=5 is wrong and how to correct it. Instead, with the work you did for them, they only need to verify/accept it. Furthermore, they are more likely to look closely at your proof (because they want to find an error to show that 2+2=5 is still true) than they are to rethink their own proof, which they assume to be correct. If they still say "Well, we showed 2+2=5 and thus your proof has to be wrong, so we won't bother to look at it -- find the error yourself," then you should consider pulling out from the project.

Of course this needs quite some effort from your side, so be sure that 2+2=4 is true and that you really want to publish this result (maybe consult an external expert on the field and discuss with him what 2+2 should be).

If you can't come up with a proof of 2+2=4, try to find a question. Say "I tried to apply our theorem of 2+2=5 when I tried to count these apples, but it just doesn't work out. Could you maybe look through this example with me and tell me how to properly apply the theorem?" Once again, they might be more likely to look for an error in your computation of the example than to look for an error in their proof. If they are also too stubborn to help you with that, you should really consider how to not work with them again or how to prevent the wrong result to get published.

And last but not least: Try to talk in their language. Depending on your fields, you might have rather different points of view on certain problems and questions. Maybe they are from number theory and thus 2+2=5 is a special case of some very important theorem, and you are from combinatorics and thus count 2+2=4? In this case, they are more likely to listen to you if you manage to come up with an example from their field or to at least adjust to their language and point of view.