Proof of the Hardy-Littlewood K-tuple Conjecture in the Distribution of Numbers Coprime with the Primorial

In the symmetries in the numbers that are coprime with the primorial we find proof of the existence of infinitely many twin primes and prime k-tuples. Deep analysis of the sieve's mechanisms confirms there does not exist the means for the K-tuple Conjecture to be false. We show and prove that Hardy and Littlewood's formulations of statistical predictions concerning prime k-tuples and twin primes are correct.