Consider this, you get a block, block Z. You do not have block (Z-1), so you check the skiphash and ask for block Y. You also don't have block (Y-1), so you check the skiphash again and ask for block X. Now block X is one that you do have, so you stop there. How much should you trust blocks Y and Z at this point?
...
Unless I'm wrong, of course. It could be that the skiphash chain rules add up to a much stronger context than is immediately apparent to me.
Branch and Bound! You've now established that block X is at least a common point in both chains (assuming you already have a chain, it's the genesis block if you're starting from scratch). Block Y is the largest hash in between X and Z. You can now repeat the process on either sub interval. Each time you go down a level, there are more blocks to check, and you get an increasingly precise estimate. If you keep going, eventually you will process every block.
I imagine that there would be a lite client that says "Estimated work: 1 bajillion megahashes. Likelihood: 76%". The likelihood number increases, very rapidly at first, and slowing down at 99%, 99.99% and so on. If it hits 100, then you have validated the entire chain like a full node.
This also allows you to quickly find the true fork point between two chains, even ones that have had difficulty changes along the way.
What do you do with the subinterval exactly?
You don't
know any blocks between X and Y or Y and Z. You don't know how many there are, you don't know what their difficulty is, you don't know what their values are, you don't know
anything at all about those intervals, except that if you get block Y-1 (or Z-1) and the value is higher than Y (or Z), one of the blocks is lying. But which one?
And since you don't know anything at all about these intervals, your estimated work is going to be W>X+(y*Y)+(z*Z) where y and z are
probably greater than 2. And I just don't see any way, using this method, to improve our estimates.
Again, I might be missing something, but from what I see, it looks like you are using knowledge of the blockchain in your estimates, which is a problem, since this is intended to be a system for finding out information about the blockchain that we assume that you do not actually possess.