Mathematical proof or any reference of "Deterministic random bit generator cannot produce more randomness than the randomness of seed"
Define the Mutual Information of a pair of random variables.
$$I(X; Y) = H(X) - H(X\mid Y)$$
For discrete random variables we hae that $H(X\mid X) = 0$, so:
$$I(X; X) = H(X)$$

The Data-Procesing Inequality states that for any function $f$, we have that:
$$I(f(X); f(Y)) \leq I(X; Y)$$
While we won't need it here, this includes randomized functions, provided they use randomness independent of either $X$ or $Y$.
Combining these two, we get that:

$$H(f(X)) - H(f(X)\mid f(X)) \leq H(X) - H(X\mid X)$$
If $f$ is deterministic, I believe the conditional entropies become 0, giving us:
$$H(f(X)) \leq H(X)$$
So the output distribution of a DRBG is always lower entropy than the distribution that its seed is drawn from.