Applying Noether theorems to Riemann zeta function and Sieve of Eratosthenes

We differentiate the rigorous Statistical mathematical proofs from the rigorous Non-statistical mathematical proofs. Statement 1: As faithfully present in self-dual L-function of Dirichlet eta function [that represents analytic continuation of Genus 0 curve Riemann zeta function], the entire Set Nontrivial Zeros containing infinitely many elements is only located on its Critical line. Statement 2: As faithfully computed using Sieve of Eratosthenes, the entire Set Gap n Odd Primes containing infinitely many elements is precisely constituted from Arbitrarily Large Number of Subsets Gap n = 2, 4, 6, 8, 10... Odd Primes with each Subset again containing infinitely many elements. Using correct and complete mathematical arguments, we show these two statements are true only if Noether theorems [involving Science of Symmetry and Law of Conservation] are fully complied with.