Claude can't stop itself from writing code comments

It's all just training.

LLMs are trained to reproduce what they were trained on. They are trained on tons of source code ... with comments.

It doesn't matter what you put in context, because what you put there will be in direct conflict with what the model "wants" to do (ie. what the patterns in the data it was trained on did).

This is the same reason why it is SO FUCKING HARD to make Claude shut up! Everything in it's training is focused on producing text, so it does it's damnedest to try and produce text.