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).