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. 本文系统构建一套分层加权筛计数框架,用于研究区间内孪生素 数对的渐近分布性质。经典筛法在孪生素数问题中普遍存在余项控制 不足、主项系统性偏移、高阶关联误差难以压制等缺陷。本文通过逐 层递增筛约束范围、构造模剩余类密度权重体系、引入解析校正常数, 完全修正传统筛法的固有主项偏差。 本文严格证明:在加权筛余项趋于零的标准解析假设下,孪生素 数计数函数满足经典 Hardy–Littlewood 渐近公式,从而推导出孪生 素数无穷存在性。全文完整给出 Mertens 定理展开、无穷乘积渐近化 简、主项偏差极限推导、边界余项阶估计、加权筛主项匹配证明等全 部中间步骤,无推导跳步。大尺度有限区间数值实验验证了理论渐近 趋势的一致性。 附录(含 Lean 4 形式化验证源码)