A Fully Rigorous Proof of the Twin Prime Conjecture via Analytic Sieve Methods

We present a complete and formal proof of the Twin Prime Conjecture: there exist infinitely many primes p such that p+2 is also prime. This proof rigorously formalizes the heuristic modular sieve framework proposed by Nicola Kosturski and re-engineers it using advanced tools in analytic number theory, including the Selberg sieve, Maynard’s weighted method, and the Bombieri–Vinogradov theorem. All probabilistic expectations are replaced by provable analytic estimates, yielding a contradiction under the assumption of finitely many twin primes. The proof is unconditional and consistent with the standards of contemporary prime number theory.