There is no such thing as a math proof for randomness in the nature of our existence. Math proofs are for Theorems axioms.
en.wikipedia.org
A
mathematical proof is a
deductive argument for a
mathematical statement, showing that the stated assumptions
logically guarantee the conclusion. The argument may use other previously established statements, such as
theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as
axioms,
[2][3][4] along with the accepted rules of
inference. Proofs are examples of exhaustive
deductive reasoning which establish logical certainty, to be distinguished from
empirical arguments or non-exhaustive
inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in
all possible cases. A proposition that has not been proved but is believed to be true is known as a
conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.
Good luck trying!!!