The Twin Prime Conjecture asserts the infinitude of prime pairs (p,p + 2). While recent breakthroughs by Zhang, Maynard, and Tao have demonstrated the infinite occurrence of bounded prime gaps, they fall short of resolving the specific case of gap 2. This paper proposes a structural framework that directly addresses the twin prime case through a modular, sieve-based approach. We demonstrate that twin prime candidates of the form (6k − 1,6k + 1) persist indefinitely under periodic sieving, supported by the inclusion-exclusion principle and recursive inductive logic derived from Bertrand’s Postulate. Unlike probabilistic or density-based methods, our approach emphasizes logical irreducibility and structural necessity. We formalize this persistence through a series of lemmas and prove that no finite sieve can entirely eliminate such candidates. Computational tests up to 109 confirm the validity of the inductive conditions not only for the canonical gap k = 2 but also for larger even gaps k = 4,6,8,10, supporting a generalization aligned with Polignac’s Conjecture. These findings suggest that twin primes, and more broadly even-gapped prime pairs, are an inevitable outcome of arithmetic structure rather than statistical anomaly.