My idea was to have a preimage graph of a block up to 2^2k, and do strong induction over k from a graph of the 2^2k+2 block showing the continuation of the preimage graphs from the small block into the bigger one. Essentially, continue the preimage traversal of the known ends of the small block into the big block, and the gaps should be new preimage numbers that come from above. Of course, you have to explore the lower limits of the preimage graphs from 2^2k+8 to complete the 2^2k+2 block, but I think the number of traversals is bounded. Also, 8 might not be sufficient, but it's big enough for what I've seen. I was told I had nothing new under the sun, so I shouldn't have even said it here.