> You can reference things outside of the sequence, but only if they are knowable before the sequence is generated. PI is known, being a fundamental mathematical constant, but the output of the transistor is not.
PI is just the limit of a sequence. It happens to be particularly useful in Mathematical theorems, but is it any less random because of that? Which of the following is not a random number, but a useful mathematical constant?
The number being recognizable is not a sufficient definition. Its like the old joke about the mathematician and the license plate. "KZJ-1249" what are the chances of seeing that! Every real number is special in its own way, but I agree the process is more important than the actual output. One wants to compare multiple instances of the same RNG process, recognizing that a true RNG almost never (but not never) outputs a sequence that is "obviously not random."
> I think I see your problem. Without time, randomness is indeed impossible.
In a some fundamental sense that is true. The larger point is that any deterministic system is "without time" in this sense, and that contrary to commonly held views Classical Mechanics is not deterministic, and QM can be deterministic.
So when people say QM is random, mechanics is not, they are usually assuming a non-deterministic interpretation of QM, which is not required by experimental evidence (only non-locality is required).
> the question is whether the rules of the system permit _anyone_ to predict the next number with 100% certainty, given all the information which it is possible for them to know
This is perhaps the closest to a meaningful definition of "random," but it depends upon the validity of the uncertainty principle, and who qualifies as _anyone_. It is in fact what I said earlier, that QM is how we define "random", not that QM *is* random.
In particular since physical law can never be proven complete and correct, "random" can never be proven to be correctly defined. QM is our best understanding of the rules of the system, and thus our current best standard for "randomness." If we find the rules to be wrong, and that the uncertainty principle is not fundamental to future models of physics then our definition of "random" will change, and something else might be brought forward as the standard for "randomness."
In this respect saying "QM is random" is vacuous, and is the same as saying "QM is our best understanding of physics." Before we had QM we had Brownian motion which could be have been appropriately called "random" although we now call it merely "chaotic."