Polynomial Time Convergence for NP-Complete Problems via Bounded Carry Algebra: A Hierarchical Reduction Algorithm for the Subset Sum Problem

This paper presents a deterministic, polynomial-time algorithm (Hierarchical Carry Reduction: HCR) for the Subset Sum Problem, a classic NP-complete problem. By mapping integer sets into a vector space and analyzing carry transitions across hierarchical layers, we demonstrate that the number of active states is strictly bounded by O(n).本文提出了一种针对经典NP完全问题子集和问题（Subset Sum Problem）的确定性多项式时间算法——分层进位约简（Hierarchical Carry Reduction，HCR）。通过将整数集合映射至向量空间，并分析分层结构中的进位传递过程，本文证明了活跃状态的数量严格受限于O(n)复杂度。