Lets take the Q function above as φ, and α as the conditional probability between y and x.
Since conditional probabilities can multiply, and each such element is the output of φn(y), then we see the rules of closure applying.
We also see turing contrasting y (left-recursive) terms of the input with x, where we should think of H1 contributing…(y) and H …contributing (x).
Now it makes sense that “the domain of definition” of φ is H1, thinking of it as an expression of the subgroup (formed from its generators) then to be conjugated by y or x (selecting a coset). (Hmm, this doesn’t feel quite right, but its along these lines.) That is, we can divide y up into subsequences however we wish, so long the elements are members of the H1 coset. We have a generative representation of y, that is, in terms of the subgroup.
Now don’t we recall from the first half of the manuscript that he gave an example of calculating a word-norm. There he conjugated a given upright, and then calculated its invariant. Could we see φ as being similar, in we get a coset within a subgroup of H1 (selected/conjugated by y)
After all, Turing went to quite some trouble to show how, when searching for non-exceptional groups, how the invariants were applied.
Considering that in the limit of this operator sequence (and thus φn) there is no uncertainty, therefore the conditional probability factor that y contributes would be 1. Thus it also makes some sense that the entropy K of x would be constant for each term in the coset (i.e. 1/4 in the case of the quaternion group using I and j as generators).