We establish a universal theorem determining necessary mod-6 constraints for any prime or composite k-tuple pattern. For any pattern (n+a₁, ..., n+aₖ), simple mod-3 arithmetic on the offsets determines whether the starting element must satisfy n ≡ 1 (mod 6), n ≡ 5 (mod 6), or both. We prove a dual theory for composites and verify the theorem on 24 patterns from pairs through nonuplets with 100% accuracy across 51,000+ instances in 100 million primes and 100 million composites.
J. M. Keen (Mon,) studied this question.