The Logarithmic Collapse of Primorial Sieves and a Proof of the Twin Prime Conjecture

Abstract This paper provides a formal proof of the Twin Prime Conjecture by establishing a definitive lower bound for the counting function \Lambda(k) within the quadratic intervals [p_{k},p_{k+1}^{2}]. By leveraging the algebraic periodicity of the primorial p_{k}\# and the Chinese Remainder Theorem, we derive an exact inclusion-exclusion expression for the count of prime pairs. We demonstrate a phenomenon termed "Primorial Sieve Collapse," where the sifting complexity truncates logarithmically at a threshold m=\lceil2\ln(p_{k+1})\rceil. This collapse ensures that cumulative discretization errors grow at a sub-exponential rate O(k^{\ln k}), which is fundamentally dominated by the quadratic expansion of the search interval and the exponential primorial signal O(k^{2}e^{k}). We prove that there exists a threshold constant N such that for all even integers k\geq N, \Lambda(k)>0, thereby guaranteeing the existence of an infinite sequence of twin prime pairs as k\to\infty.