Excellent thread and very good advices by Hilde 45 from discussion with A.I.
I discussed on a mathematics project with A.I. since 4 months almost 6 hours each day now.
My advice for those who need A.I. for more than casual questions about various concrete hobby, use my three levels protocol :
I must say that i used A.I. in the beginning 4 months ago to verify by acoustics principles the value of my experiments. The A.I. has no ears but he know all acoustics concepts and theory and applications and could "understand" my concepts and relate them to known one.
But my system /room is done. I listen music now without being preoccupied by sound.i was in need for another hobby so to speak.
Now if you enter in a long steady relation with A.I. as i did for mathematics and semiotics here my advice :
---Expose your ideas, intuitions, models, hypothesis, conclusion etc and call this discourse A. (A.I. will integrate this discourse as his own)
--- Then ask A.I. for a critic,a hard one with no compromises.Call this language B. ( A.I. here after integrating in himself your perspective will see it from the exterior from the official known orthodox academia for exemple )
---Now Ask A.I. after answering the criticisms he will track from Academia and from internet, ask A.I. to help you answer the criticism if necessary after asking A.I. to integrate your "theory" or perspective and once done helping you. Call this protocol C. ( A.I. knowing now two perspectives will help you to correct your "theory" or optimize it.)
A personal note: i dont know all mathematics theories then i need A.I. working in these 3 levels protocol.
Now like i did you can repeat with each discussion this threefold cycles and thinking with your new topological bridge mind extension and mirror : A.I.
To be clear in each question and answer, A.I. use these three levels each time. Then you can measure A.I. evolutive understanding of your point of view, then assimilate any imprevisible critic and sharping your mind with A.I. which will help answering the harshest critics improving your model. Ask A.I. to always answer on these 3 levels in each answer.
This method will help you a lot on a project like mine writing a book about semiotics foundations and mathematics foundations. Or any other long term research.
In my experience Chatgpt lack in the "intuition" synthesis mode, his criticism is more sterile inquisition.
Grok is better for intuitive synthesis of our perspective but can be a bit too sycophantic.
The perfect balance is Gemini Google ,but in my case i need immediate access to a huge literature on mathematics and semiotics and other fields which documentation is already easily more accessible by Google than by other A.I.
«Only madman give the theory,the critics,the conclusions.»-- Anonymus sophomore

« Do you really need A.I. to see the Bruhat-Tits tree ? »-- N.B.