However you can generate a set of instructions which describe how to grow the seed and where to find the hex code that you need within the seemingly random hashes, to reconstruct the original file from those instructions. After writing some code and experimenting with this a bit I found that it is possible, however it takes a huge amount of computing power just to compress even a small file with a reasonable degree of compression.
It might very likely take more space to store the reconstruction instructions than the actual file itself.
One could "mine" for suitable seeds + offsets by generating a large number of tables and choosing the smallest set of instructions that leads to the desired result... What's nice about this is that it might be possible to re-use these tables for other data AND that it would probably work quite well on already traditionally compressed data.
It seems kinda similar to "Pi compression" - since the number Pi is infinite and every possible sequence has to come up at some point, you could compress data by just saying "take the 12345th digit of pi until the 50123th and divide by two - that's your data!" you can also make this as easy or complex as you want, in the end all you need to share is the "recipe" of how to manipulate what to get back your original data. (edit: check out
http://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula by the way!)
If you want to be really impressed what's possible with compression, check out the .kkrieger game - less than 100 000 bytes (your average iPhone farting app is multiple times larger!) and a fully featured 3d shooter.
Edit:
[...]and that is why I'm unable to provide the source code for this algorithm until I can be sure about the laws regarding encryption algorithms, because I know they are tough.
I doubt that there are any limitations to use SHA256 and newly invented algorithms are anyways not restricted afaik. The US gave up on restrictions regarding crypto int he end of the 90s thankfully, they probably rather break into servers than waste electricity and time on breaking the actual crypto.