Ok, started reading through the whitepaper, and have some questions. Before I raise them however, given this is a new consensus protocol, it might be best to present a simple example using imaginary coins and people, so that we can all visualize how this works.
Now the questions:
1. If I understand this correctly, every PoP miner would need to transact on the BTC network and embed in their transaction the entire block header of the SI chain, which is being mined by the PoW miners. The two issues I see are a) You are asking PoP miners to pay BTC transaction fees, b) It becomes a race of who does it first so some PoP miners might opt to pay higher fees for their transaction to be mined first and c) The financial incentive would need to be significant for a PoP miner to do so, and cover the BTC fees
2. While you are saying that it would not be practical for someone who forks the BTC chain to take over (re-write) the SI chain, considering that the SI chain would probably be made up of a small amount of hashing power (otherwise, why would you need the hashing power of BTC), and there would already be large pools in play, all they would need to do is also fork the SI chain and provide a solution for the next block faster.
Generally, chapter 6 which deals with #2 above reads very weak, there seems to be some wishful thinking and statements along the lines of "As such, anyone watching the SP blockchain would see what block(s) are at risk for the fork, how much stronger (or weaker) the current chain is compared to the adversarial partys chain, and could potentially use some means (like balance-based voting) to invalidate the adversarial chain before it is released to the network." don't inspire confidence.
It is ok if you haven't figured everything out yet, be upfront and work with the community. Right now this seems like an overcomplicated solution looking for problem though.
Don't mean to sound critical, and apologies if I have misunderstood anything.