I believe it makes it more deterministic, but not much more deterministic. It knows the entropy budget and only consumes a small amount.
I do not believe that it is theoretically necessary for it to be more deterministic at all. If it were deterministic, it would be recording not just the watermark that it is produced by LLM, but also the fact that it was produced at the beginning of the session, which is not the goal and thus wasteful. The purpose is that the watermark can survive the prefix being truncated, so there is a potentially arbitrary context that the watermark should not be sensitive to. It is about correlations between successive calls to the RNG, but it could still be initialized randomly. That initialization is like the unknown context.
You seem to ask: if Rome had technological progress, why didn’t it have an industrial revolution?
I reject the premise. It appears to me that in about 150BC Rome destroyed Hellenistic civilization and not only produced a backwards shock to technology, but started 1000 years of regressing frontiers.
Why focus on the point event of the industrial revolution? If we have a grasp on general technological progress, we should study the broader question. If we don’t, we might focus on the industrial revolution as an undeniable event. But then we should be cautious about claiming that Rome made progress. You mention the Antikythera mechanism as a reason for uncertainty. But this was pre-Roman. If anything, it suggests that the Romans were behind the Greeks. You mention Archimedes, who was also pre-Roman. You mention Vitruvius, who was Roman, but was fairly early. In particular, you can see regression by comparing him to Frontinus, a century later.
You say that Rome had centuries of per capita growth. But your link is specifically about Britain. That was the most backwards Roman province. It did see centuries of technological advance. Rome did spread technology even as the height fell.
Why did Rome see technological regression? Actually, I think that is not such a difficult question. Rome had insecure property rights leading to consolidation and diseconomies of scale.
Of course, this leaves me with the question: Why didn’t the Hellenistic Egypt have an industrial revolution?