Some great thoughts, many of which extend beyond mathematics.

One in particular, about sharing secrets, that had not crossed my mind (I have found that virtually no one ticks the "do not train on my data" box...):

> Similarly, it seems that mathematicians already concerned and guarded about what they write in research statements and research proposals. After all, an administrator or a reviewer may upload those documents into an LLM to get help in writing a report, thus potentially compromising the confidentiality of unpublished research.

>One in particular, about sharing secrets, that had not crossed my mind (I have found that virtually no one ticks the "do not train on my data" box...):

When said button is in the basement, at the bottom of a broken set of stairs, locked behind a door that says "beware of the leopard", inside a filing cabinet, and implemented as

    dontTrain() { //NYI
    }

That's a cultural artifact. Culture comes and goes. Read about how hiding mathematical discoveries was commonplace in Fermat's time and before.