We present a deterministic primality sieve derived from the multiplicative structure of integers modulo 30. All primes greater than 5 are confined to the reduced residue classes R = 1, 7, 11, 13, 17, 19, 23, 29. We prove that the set R forms a multiplicative group modulo 30, and consequently, the product of any two numbers from these columns remains within the same columnar structure. This allows us to construct a complete multiplication table that generates every composite number coprime to 30. Our central theorem states that for any fixed column c ∈ R, the numbers in that column which are not generated by the multiplication table are exactly the prime numbers. Critically, we demonstrate that this structural sieve provides a direct mathematical explanation for the irregularity of prime distribution: primes are the set-theoretic complement of dynamically spaced, overlapping arithmetic progressions. Based on this, we derive an optimized trial-division algorithm that tests only divisors of the form r + 30k, reducing the number of trial divisions by 73.3% compared to naive trial division and 46.7% compared to odd-number-only division.