Thank you for the example. I have two initial comments and possibly more if you are interested. 1. In all of the literature on the problem, the sequences that we compare specify social states. When we compare x=(x_1,x_2,...) and y=(y_1,y_2,...) (or, as in your example, x=(....,x_0,x_1,x_2,...) and y=(...,y_0,y_1,y_2,...)), we are doing it with the interpretation x_t and y_t give the utility of the same individual/generation in the two possible social states. For the two sequences in your example, it does not seem to be the case that x_t and y_t give the utility of the same individual in two possible states. Rather, it seems that we are re-indexing the individuals. 2. I agree that moral preferences should generally be invariant to re-indexing, at least in a spatial context (as opposed to an intertermporal context). Let us therefore modify your example so that we have specified utilities x_t,y_t, where t ranges over the integers and x_t and y_t represent the utilities of people located at positions on a doubly infinite line. Now I agree that an ethical preference relation should be invariant under some (and possibly all) infinite permutations IF the permutation is performed to both sequences. But it is hard to give an argument for why we should have invariance under general permutations of only one stream.
The example is still unsatisfactory for two reasons. (i) since we are talking about intergenerational equity, the t in x_t should be time, not points in space where individuals live at the same time: it is not clear that the two cases are equivalent. (They may in fact be very different.) (ii) in almost all of the literature (in particular, in all three references in the original post), we consider one-sided sequences, indexed by time starting today and to the infinite future. Are you aware of example in this context?
I am curious about your definitions: intergenerational equity and finite intergenerational equity. I am aware of that some literature suggests that finite permutations are not enough to ensure equity among an infinite number of generations. The quality of the argumentation in this literature is often not so good. Do you have a reference that gives a convincing argument for why your notion of intergenerational equity is appropriate and/or desirable? I hope this does not sound like I am questioning whether your definition is consistent with the literature: I am only asking out of interest.