Conditional Reduction Framework for the Twin Prime Conjecture: Weighted Sieve and Residue‑Class Solvability
Year: 2026
Type: preprint
Abstract: The twin prime conjecture is a sub‑problem of Hilbert’s eighth problem. It asserts that there exist infinitely many primes p such that p + 2 is also prime. The Brun sieve and the bounded‑gap prime theories of Zhang–Maynard yield infinitely many prime pairs with bounded gaps, but cannot reduce the gap exactly to 2. Traditional sieve methods are obstructed by high‑order remainder barriers; one cannot logically rule out the possibility that twin‑prime counting vanishes for all sufficiently large intervals. We carry out analysis from two perspectives: at the modular‑arithmetic level, we discuss solvability of congruence systems for any prime modulus q. At the ana lytic‑sieve level, we construct a weighted sieve tailored for gap‑2 prime pairs, aiming to reduce the classical sieve remainder o ( x/ log2 x ) to a bounded infinitesimal o(1). We establish a conditional reduction: if the weighted‑sieve remainder‑nullification conjecture holds, then the twin prime conjecture follows. This paper only presents a logical reduction skeleton; core analytic statements are listed as open conjectures. We compare remainder orders of classical Brun and Selberg sieves, discuss construc tion motivation of the weight function, and point out intrinsic obstacles within the framework. Lean‑4 formalization source code is provided in the appendix. This archive presents a conditional reduction framework, not an unconditional proof of the twin prime conjecture. Appendix contains Lean‑4 formalization skeleton source code. 孪生素数猜想为希尔伯特第八问题的子命题,该猜想断言存在无穷多素数 p 使 得 p + 2 同样为素数。布朗筛、张益唐–梅纳德有界素数间隔理论可以证明存在无穷 多具有有界间隔的素数对,但无法将间隔压缩至 2。传统筛法受高阶余项壁垒约束, 不能严格排除充分大区间内孪生素数计数整体消失的可能性。 本文从两个角度展开分析:模算术层面,对任意素数模 q,讨论同余方程组的 合法剩余解存在性;解析筛层面,构造适配间隔 2 的加权筛函数,将经典筛法余项 o ( x/ log2 x ) 压制为有界无穷小 o(1)。本文建立条件归约:若加权筛余项零化猜想成 立,则孪生素数猜想成立。本文仅给出逻辑归约骨架,核心解析引理作为未解决猜 想列出,同时提供 Lean4 形式化声明源码。文中详细对比经典 Brun 筛、Selberg 筛 的余项阶数,解释现有筛法体系的理论,讨论权重函数的构造动机与猜想内在障碍。 本归档为条件归约框架,不构成孪生素数猜想的无条件证明,附录包含 Lean‑4 形式化骨架源码。
Author: Wei Changmin
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7207718507
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