So i've read from various places that maximal compression produces a truly random output. But suppose you were going to apply compression to a truly random series of numbers with something like LZW. Well wouldn't it be the case that since randomness often produces sequences which do not appear to be random, that therefore compression would work at some point in a random sequence of numbers? In the case of finite sequences it means that the efficiency of LZW here would be random since a true random number generator could produce a compressible sequence of numbers. Thus true randomness does not mean absolute compression. In an infinite sequence of random numbers you would expect it to contain many sequences which don't actually seem random like how randomness could produce the sequence 111111111111 purely by chance. Its similar to how in PI you can find sequences of numbers which don't appear to be random, but in the context of a larger set of numbers the distribution between all different numbers is the same. So clearly this is compressible. This implies that an infinite sequence of numbers is infinitely compressible in theory right? So really absolute compression != true randomness because it only works in some cases where a finite sequence of numbers was random and has no sequence or pattern of numbers. From my understanding of randomness, it would be impossible, in an infinite sequence of numbers, to find no compressible patterns at any point. I don't have a proof for this but if anyone knows of one or can show me how i'm wrong i'd love to hear it. Thanks.