"Suppose these are truly fair, truly random dice." - Wait a minute. You are smuggling a very strong assumption into your thought experiment here, which is precisely the whole point of my question. First of all, examples involving dice and similar objects are problematic for the reasons I recently added to the OP (take a look). Secondly, how can truly random dice be fair? That seems question-begging, which is exactly what my question is asking in the first place. How can something truly random be forced to follow a uniform (i.e., "fair") distribution?

@user80226: if the dice aren't fair, you will get a different probability distribution, precisely because that is not complete randomness. Probability predictions are based on explicitly stated assumptions; change the assumptions and you change the prediction. If you are asking how we distinguish whether something is truly random, that's like asking whether a statement is ultimately true; the only way to find out is to keep running the experiment until you are adequately convinced, and even then you can be wrong. In theory, practice follows theory, but in practice...

@keshlam "if the dice aren't fair, you will get a different probability distribution" - If the dice in your thought experiment are obeying rigid body dynamics, the thought experiment is then off-topic (see the recent edit to the OP to understand why).

@keshlam I guess you mean "no more improbable"? Yes, all of that exactly.

@user80226 The thought experiment doesn't change if the dice aren't fair, only the numbers (perhaps 7 could be five or sixteen times as likely as 12, with unfair dice). Either way, it is the numbers you get on the two dice, and the rule for combining those into one outcome, which determines the result - not some kind of hidden state remembered by the system of two dice which is somehow separate to the individual dice. And if no "keeping track" is required for the system of two dice to obey a probability distribution, why should any "keeping track" be required for the individual dice?

@kaya3 You are once again talking about dice. Dice obey rigid body dynamics. That's determinism, not randomness.

@kaya: Yes. I'll repost a corrected version since that's the only way to fix the typo.

(Tupo foxed...) Whatever the dynamics are, rigid body or direction of emission of a photon or time between detection of cosmic rays,, probability depends on the specific case being considered. And don't forget that a hundred 6's in a row from a 1-6 random number generator of any sort is still a valid random result, no more improbable than any other specific sequence of 100 random 1-6 results. If you are looking at totals, or unordered sets, those are different cases with their own probability distributions.

I would also quibble with the assertion that physical objects are deterministic in a sense that is meaningful here, but that is a completely separate discussion and should be dropped out at this one.

@user80226 It doesn't matter where your random inputs come from. If you don't believe in dice, add the radioactive decay times of two unstable nuclei, or whatever else you prefer. You are picking at things which are not relevant to the point.

I think the part you are hung up on is the idea that probability distributions must be manifested in physical objects, such as dice. The answer is that they don't need to be, because "random processes" are not the same thing as "probability distributions". As the last part of my answer states, "It also doesn't have to be physically possible to take independent samples from a distribution."

When you don't abstract away the usual physical details, it may well be the case that repeated rolls of a real, non-ideal, physical die are not independent or identically distributed. Nonetheless, that doesn't refute the concept of a probability distribution, because a probability distribution is an abstract mathematical object, not something which exists in reality. Just as the number 17 exists in the ontology of mathematics, so does the uniform discrete distribution on six elements. How can we know a physical die exhibits that distribution? Only by statistical tests and inductive reasoning.

But even if a real, non-ideal, physical die does not roll independent, identically distributed outcomes, there is still a probability distribution associated with each separate roll of that die. It's just that the distributions for each separate roll might be different, and we don't necessarily know what those distributions are, and it might not be physically possible to sample any individual one of those distributions more than once (if doing so somehow changes the physical state of the die, or the universe in which the die is rolled). But they are distributions nonetheless.

@kaya3 Your answer proposes a deterministic protocol/function F(X), where X is some noisy input. If X is deterministic, the whole thing is deterministic, and so it's not relevant to my question. Dice following deterministic mechanical rules would be an example of a deterministic X that is not relevant to my question. If X is truly random, then F(X) would be a deterministic transformation of truly random input. However, I don't see how F(X) would itself be random following a probability distribution unless X itself already follows a probability distribution.

But if you assume that, then that shifts the problem to now having to explain why X follows a probability distribution in the first place. That's why I don't see how your answer can solve the problem. It only shifts the problem to the inputs. Why do the inputs follow a probability distribution in the first place?

@user80226 The point is to show that distributions don't need anything "kept track of" in order to be followed. A system of two dice obeys a distribution which neither die individually obeys, neither die is responsible for or capable of "keeping track" of what that distribution is supposed to produce in the long run, and a human adding the result of the two dice in their head doesn't magically create some other physical entity which "keeps track" of the dice together to ensure the distribution is obeyed in the long run. So a distribution is followed without any "keeping track".

@kaya3 But that's question begging, because you haven't explained yet how your inputs are able to be (1) truly random and (2) followers of a probability distribution in the first place without keeping track of information. Assuming that for your inputs in order to explain how something else doesn't need that is question-begging with respect to the inputs.

It is not question begging, you are just reading my words as if they are an argument for a different conclusion than what I have stated. It doesn't matter how the individual dice are able to obey a distribution; so long as you accept that the distribution of the sum of the two dice is different to the distributions of the individual dice, then there is a new distribution with no physical entity keeping track of that distribution. Even if you believe the individual dice "keep track" of their own distributions, what physical entity keeps track of the distribution of their sum?

Let us continue this discussion in chat.