I also appreciate Glen bringing in Lovasz representations. I suspect we run into a similar problem there: if two people are random walking through the same sufficiently mixing space, their long-run statistics become indistinguishable.
What I keep picturing are random walks and their space-filling quality. As more of the walks accumulate, we have to zoom out to see everything, at the expense of the finest details. Eventually it saturates as a consequence of all representations living on finite hardware. What began as two distinct histories becomes one large and smooth sphere of noise—not simply because we stopped looking closely, but because the accumulated representation no longer retains what distinguished them. I suppose I am also thinking about LLMs diligently interpolating the convex hull of the human-written archive. Under additional conditions of *forgetting*, two originally distinct sequences can become the same for an arbitrarily long time.
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