@modulux A proof is not information in a strict sense, and largery exactly because of this reason: it's self-contained (or, well, can be, with sufficient formalism available).

In a broad sense, there's some very interesting philosophy that can be done about the notion of information content of Teh Book. But it's mostly the kind of philosophy that requires a larger mug of beer than would be conducive to my upcoming meetings[1], so, as the old Orcish saying goes, nar udautas.

As a general rule, I tend to prefer the interpretation that a proof is a series of "I'd now like to bring your attention to ..." kind of steps: they don't add anything (directly) to your mental map; they suggest where you should look at to find interesting things that are already on the map.

[1] A children's book I once read included a character, one mathematics professor, who argued that it is pointless to ask questions, because there's two possibilities: the answer either is known or is not known. If it's known, what's the point of asking it again? If it's not known, what's the point of asking if there won't be an answer?

And, well, while it's silly in an obvious way, this kind of reasoning actually comes up in the context of proofs-as-information.

@proedie No, that's not how information works. Information is about reducing your uncertainty space. Every time you can exclude half of the uncertainty space, you will have gained one bit of information. If you exclude less than half of the uncertainty space, you will have gained less than a bit of information. Just ask Claude[1].

Looking at broken clock[2] does not reduce your uncertainty space at all, therefore you gain zero bits of information. The classic formula Claude Shannon is famous for involves dividing the volume of the uncertainty space after gaining information with the volume of the uncertainty space before gaining information, and then taking a base-2 logarithm of the ratio and negating it. If you don't care a minus one bit about negative amounts of data, you can turn the ratio on its top; then, negation won't be necessary. But there's didactic reasons for presenting it in the classic way.

[1] Claude Shannon, an overall smart human and a measurer of the enthropy of information. Who were you thinking about?
[2] Well, there's the minor issue of knowing that the clock is broken, lest you erroneously throw out parts of your uncertainty space that might actually be valid. But the problem of information-resembling text is also an issue that applies to chatbots.