The LCR insures no double-spend exist. There is no fix for the idea of taking the conjunction of chains. The idea is flawed that is why I abandoned it. Also note I had added to my prior post.
That's why you'd need a new chain selection rule... what happens if you were to use to greatest set of all new transactions as the rule?
The greatest set of all new transactions would have the largest cumulative POW weight (in a system with a constant POW cost per transaction), such that any double spend by the attacker within this set would become the canonical spend, not the double spend.
Yes I remember now that was my next line of thinking too. If the rule is that the union is the canonical set, then the attacker's latter double-spend is the invalid transaction. Remember I wrote that in my vaporcoin thread last month.
But the flaw remains that there is no way to prove what the union is. At any given time the honest chain has all the transactions from the attackers chain plus censored transactions, but then the attackers chain releases a new block with more new transactions. How do we prove which was first? That is the entire point of a longest chain rule is we have no way to prove relative order otherwise. This is what I wrote several posts before. You will just chase your tail in circles. It violates the CAP theorem.