> wouldn't a TRNG require infinite memory, ... etc.? One number generated may be 2 but the next number may be negative infinity.
I may be misunderstanding you, but true randomness doesn't mean 'a distribution that's non-zero over the whole integers' - those are orthogonal concepts. You can have a TRNG that gives you a random choice from just {0,1} - the size of the distribution isn't what makes it a TRNG, that just determines how much entropy you need.
(And a uniform distribution over the integers is impossibly no matter how good your RNG is, as it's mathematically undefinable).
I may be misunderstanding you, but true randomness doesn't mean 'a distribution that's non-zero over the whole integers' - those are orthogonal concepts. You can have a TRNG that gives you a random choice from just {0,1} - the size of the distribution isn't what makes it a TRNG, that just determines how much entropy you need.
(And a uniform distribution over the integers is impossibly no matter how good your RNG is, as it's mathematically undefinable).