Residue + Linear Correction Model for Twin Prime Distribution

This work presents a dual contribution to the twin prime problem. First, it proposes a refined model for estimating the distribution of twin primes π₂(x), based on residue-lattice exclusion and adjusted by a linear correction term. The model closely matches known values up to x = 500,000 and demonstrates strong empirical stability. Second, it provides a theoretical proof of the infinitude of twin primes through a structural approach rooted in modular admissibility and density preservation. By showing that the sieving of twin candidates retains a positive density asymptotically, the argument concludes by contradiction that infinitely many twin primes must exist. Together, these analytical and structural results unify predictive modeling with rigorous number-theoretic reasoning, offering a coherent response to one of the most enduring questions in prime number theory.