I think you missunderstand. If the llm can "add back" 200 bits of information, these 200 bits are superfluous by definition.
They can quiet literally be inferred from the starting 300 bits. And the llm has proved it.
If the receiver wanted these 200 added bits, she could infer them either herself or even use an llm to do it.
In principle, all of mathematics can be inferred from the axioms. The field of mathematics is not superfluous and something a receiver could infer themselves if they wanted it.
My brain does not contain all the information that can be added by an LLM; a human brain could contain it, even the biggest LLMs are about 1% of the (if you approximate synaptic count ~= parameters) parameter count of a human brain, but none actually will.
What my brain may actually contain is the information necessary to verify the (in this example) more than 200 bits the LLM claims to have added and trim out the parts which are false, retaining the (in this example) 200 "new" bits of new information added by the LLM.*
Concrete example: I am a software developer by training, though not a web developer. If someone who does not have any developer experience asks me to make a web app, I am forced to use an LLM as I do not know enough JS etc syntax to get it done myself. But as we all know, LLMs are only "ok" but not "good" at making software, so there are a lot of rough edges and outright mistakes. My experience as a software developer extends to detecting such failures and I can usually correct them.
The original person, someone who has no developer experience, can also prompt the LLM. Right now, this would result in something that retains all the errors, because they didn't have someone like me intermediating between them and the LLM.
I add bits by removing noise, the LLM adds bits but they contain noise.
I do not know for how long this will remain true, but today it is true.
* Feels like P versus NP to me. The answers AI generate are at their best when they're easy to verify. Then again, when they're easy to verify, they can be RLed to get good at this quickly and the need to verify goes down, leaving them still pretty bad at things that are hard to verify.
> If the llm can "add back" 200 bits of information, these 200 bits are superfluous by definition.
No, they're not. Getting those bits takes energy.
SOTA LLMs know way more than any individual on approximately anything there is to know (and what they don't, they can look up faster than people can). It's very easy for them to make the "missing" 200 bits explicit, rather than implicit, which in practical terms is the same as adding 200 bits that weren't there before.
Theoretically, an idealized omnipotent mind / AGI could derive the unifying theory from reading your HN comment on a phone screen. There is enough information there, if you were able to extract every bit of evidence available from it. But you are not. Neither am I. It would take us practically infinite work to try, solving this most cruel mathematical riddle.
If the receiver wanted these 200 added bits, she could infer them either herself or even use an llm to do it.