Layered Weighted Sieve and the Twin Prime Problem (with Lean 4 Formalization)

This paper systematically constructs a layered weighted sieve counting framework for studying the asymptotic distribution of twin prime pairs in intervals. Classical sieve methods for the twin prime problem suffer from insufficient remainder control, systematic deviation in the main term, and difficulty in suppressing higher-order correlation errors. By successively ex panding the sieve range, constructing a modular residue-class density weight system, and introducing analytic correction constants, we completely elim inate the inherent main-term deviation of traditional sieves. We rigorously prove that, under the standard analytic assumption that the weighted sieve remainder tends to zero, the twin prime counting function sat isfies the classical Hardy–Littlewood asymptotic formula, thereby establish ing the infinitude of twin primes. The full derivation includes Mertens the orem expansion, infinite product asymptotic simplification, main-term de viation limit derivation, boundary remainder order estimates, and weighted sieve main-term matching proof, with no steps omitted. Large-scale numer ical experiments verify the consistency of the theoretical asymptotic behav ior. All key lemmas and the main theorem have been formalized in the Lean 4 proof assistant, with the complete source code provided in the appendix.