Interval Inclusion Problems for 2-Midsets of Prime Sets: Structural Analysis and Numerical Verification from an Additive Combinatorics Perspective

Abstract (English): This paper re-examines the finite truncation problem corresponding to Goldbach's conjecture from the perspective of Additive Combinatorics. By introducing the concepts of Sumset and 2-Midset, the original problem is precisely formulated as follows: find an irreducible minimal subset of primes S such that the target interval I_A is contained in its 2-Midset M(S). We explore the structural constraints of this family of minimal covering sets, deriving the core primes ({2,3,5,7}) that must be included to cover the interval when A≥6, and analyze the necessity of extreme boundary pairings in conjunction with the boundary pairing constraint. Based on this, an ascending greedy elimination algorithm is designed for numerical verification. This paper is a theoretical exploration at the intersection of additive combinatorics and heuristic algorithms. The conclusions aim to reveal the structural characteristics of prime sets under averaging operations and do not constitute a global proof of the original conjecture. 中文摘要: 本文从加法组合学的视角,重新审视哥德巴赫猜想对应的有限截断衍生问题。通过引入和集与 2-平均集的概念,将原问题精确表述为:寻找不可约的极小素数子集 S,使得目标区间 I_A 包含于其 2-平均集 M(S) 中。本文探讨了该族极小覆盖集的结构性约束,推导了当 A≥6 时覆盖 I_A 必须包含的底层素数核心({2,3,5,7}),并结合边界配对约束分析了极端配对的必要性。在此基础上,设计了升序贪心剔除算法进行数值验证。本文属于加法组合与启发式算法交叉的理论探索,所得结论旨在揭示素数集在平均运算下的局部结构特征,不构成对原猜想的全局性证明。 Note: The main text is provided in both Chinese (Prime_Midset_Cover_CN.pdf) and English (Prime_Midset_Cover_EN.pdf), with the complete LaTeX source code included.