Superb++.

I did a differential analysis, subtracting the numerical value of successive binary strings that met the criteria, and started to see a pattern emerge that seemed to be keyed to 2**N where N was the number of 0s or 1s depending upon whether iterating up or down.

But there was always some odd adjustments required at the start and end of each run which I couldn't tie down the pattern to. Then the OP started talking about infinite compression algorithms and I lost interest :)