>Of course I am slightly cheating here, but the cheat is conceptually interesting (and fun). The idea is that paradoxical compression might have only a small de

>Of course I am slightly cheating here, but the cheat is conceptually interesting (and fun). The idea is that paradoxical compression might have only a small density of problematic cases (inputs where the algorithm misbehaves, e.g. by expanding the length or not decompressing to the original input) and tools from the field of cryptography can be used to "push away" these cases so that they mathematically exist but nobody can find one with nonnegligible probability. In essence, this is transforming the question from "is it mathematically possible?" to "can I claim to do it and not being detected as a fraud?". This change of model makes paradoxical compression possible. Isn't this trivially wrong? If you have a bitstring s and you compress it to c, where the length of s is larger than length of c, compressing c again produces an issue. For example, if you compress a bitstring to just 1, how do you compress 1? To 0? And how do you compress 0 then? We can use this principle to find uncompressable bitstrings of any length anywhere for any algorithm. An idea pops into my head regarding this, though: in signal analysis there's a concept called noise shaping (which matches OPs ideas to a great degree), the idea is to push any noise out from some band and into another, it's super neat actually. Could this idea be expanded such that there is some singular monster string that expands to an absurd size, while others stay the same or are smaller? How about multiple but still small in density? My intuition sadly says no because of what I started this comment with. It's trivial to find strings that aren't compressible. Please let me know if and where my ideas on this go wrong!