The Observer establishes some equivalences and these equivalences can be updated as so-called completion rules, i.e. rules that are not really part of the evolution of the universe, but a kind of illusory rules that the observer establishes and that allow him to effectively jump from one branch of history to another, i.e. they are a kind of two-way rules that allow him to jump between the two branches that intertwine the branches of history.
I (Jonathan Gorard) mean, if the observer is described by the same computational rules as the system itself, then it's not just that the universe is branching and merging, but the observer is branching and merging too – their internal perception of the world is branching and merging at the same time.
These completion rules are a way of formally encoding the statement that, to the observer, these branches of the history are the same and they cannot tell the difference.
Well, I was able to show that if you add enough of these completion rules to the system, you can turn any Multiway System into a causal invariant system.
In other words, that any non-causal invariant multi-way system can be reduced to a system that is observationally equivalent to a causal invariant system, as long as the observer applies a sufficient degree of coarse-graining and enough of these completion rules.
See Technical Debt for a way to apply such rules.