A finite model selecting ordered slices

Conservation of this named observable is compatible with information acquisition and failure of universal recovery. It is not conservation of an unspecified physical energy. Normalized postselected branches need not preserve its unconditional input mean.

For a length-$n$ active history, all branch matrices commute. On a sector $P_j$, the outcome probabilities are

Thus diagonal inputs give a classical chain with fixed hidden sector $j$. General inputs give the mixture of the two sector history laws, weighted by their initial populations. The outcomes are independent conditional on $j$, not generally after mixing the sectors.