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

This paper provides a formal proof of the Twin Prime Conjecture by establishing a definitive lower bound for the counting function Λ(k) within the quadratic intervals [pₖ,pₖ₊₁²]. By leveraging the algebraic periodicity of the primorial pₖ# 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=2ln(pₖ₊₁). This collapse ensures that cumulative discretization errors grow at a sub-exponential rate O(kln k), which is fundamentally dominated by the quadratic expansion of the search interval and the exponential primorial signal O(k²eᵏ). We prove that there exists a threshold constant N such that for all even integers k≥ N, Λ(k)>0, thereby guaranteeing the existence of an infinite sequence of twin prime pairs as k→∞.