> There is a theorem that (...) multiplication requires unbounded memory

What theorem is that?

The multiplication of any two integers below a certain size (called "words") fits in a "double word" and the naive multiplication algorithm needs to store the inputs, an accumulator and at most another temporary for a grand total of 6*word_size

Sure, you can technically "stream" carry-addition (which is obvious from the way adders are chained in ALU-101) and thus in a strict sense addition is O(1) memory but towards your final point:

> Theory is important, but engineering is also important.

In practice, addition requires unbounded memory as well (the inputs). And it's definitely compute-unbounded, if your inputs are unbounded.

I dislike the term "we muddle along". IEEE 754 has well specified error bars and cases, and so does all good data science. LLMs do not, or at least they do not expose them to the end user

So then, how exactly do we go about proving that the result of chaining prompts is within a controllable margin of error of the intended result? Because despite all the specs, numerical stability is the reason people don't write their own LAPACK.

But it's not like these systems make theory go away, they make compromises. So the question is, what's the compromise required for an algorithm that can check the conformance of computer programs to natural language specifications that doesn't involve hoping for the best?