The pigeonhole principle means that compression can never be effective on average. For every input that can be compressed, there must be others that are expanded, such that the average compression ratio is at best 1.

Useful compression algorithms take advantage of the fact that inputs aren't evenly distributed. Real-world data tends to have repetition, self-similarity, or predictability, which real-world algorithms use to produce smaller output.

It's unlikely that πfs would provide useful compression for any real data, since it would have to line up entirely by random chance.

If computation is instantaneous, then you could recurse down to a single pointer, which points to a pair of pointers, which each point to another pair, and so on until you have an arbitrary amount of your data. Since computation is instantaneous, it is instantaneous to compute this first pointer for the contents of any hard disk. And since computation is instantaneous, rebuilding the files on the disk from that pointer is instantaneous.

Your compression ratio would approach infinity.

You mean, if there was an oracle for instant pi-index lookup? All finite-length strings can be expressed as natural numbers, so the pi-index is just a mapping from that number to another, pi: N -> N, with no guarantee of any more efficiency. Over the set of all finite-length strings, I do not think this would provide an advantage. But it might be fun.

I think the offset is bigger than the actual information you want to store in most of the cases.