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Your client only gets a page of the sorting at a time. You would only be able to sort that single page, but to do an actual sorting, you’ll need more posts than just what appears on the front page.
Right, that is what I meant by “inconsistent from one page to another”. The received posts would be in an unfiltered state on each page and so the order could be very different depending on the local blocklist, probably resulting in stuff like the top of page 2 having a higher score than the bottom of page 1 pretty frequently.