A Positive Lower Bound for the Sum of Log-Reciprocal Twin Prime Products via Weighted Sieve

The twin prime conjecture asserts that there are infinitely many pairs of primes that differ by two. While recent advances have improved our understanding of bounded prime gaps, the conjecture remains unresolved. This paper refines the weighted sieve method to estimate a sum over twin prime pairs, where each term is of the form $(1\mathrm{/}p)(log(x^{\alpha }\mathrm{/}p){)}^k$. We establish a strict positive lower bound for this sum, which implies the infinitude of twin prime pairs.