Carry ‑ Structured Block Factorization A Structural Model for Integer Factorization Based on Block Decomposition and Carry Propagation

This paper introduces Carry ‑ Structured Block Factorization, a new structuralmodel for integer factorization based on block decomposition and carry propagationin long multiplication. Unlike classical number ‑ theoretic or algebraic approaches,this method treats an integer not as a purely numeric object but as a layered structure1defined by block ‑ wise products, digit constraints, and the flow of carries betweenblocks.The key insight is that carries in base ‑ 10 multiplication encode deterministic informationabout the factors themselves. We formalize this through three concepts: carrystoppers, digit positions where the carry must vanish and the block becomes structurallyfixed; carry bridges, linear constraints that uniquely determine intermediate blocks fromdistant ones; and a nonlinear block ‑ solving order (1 ! 3 ! 2 ! 4 ! 6 ! 5 ! 7 ! 8)that minimizes dependency cycles and enables reverse determination of unknown blocks.Using these mechanisms, the factorization of large integers becomes a structuralreconstruction problem rather than a numerical search. A detailed worked exampledemonstrates how the method reconstructs factors such as A = 8921 7098 : : : andB = 1146 9802 : : : from the product N = 10223466 : : :, with carries providing the decisiveconstraints. The framework generalizes naturally: any digit position can act asa stopper, making the method applicable to integers of arbitrary size, including thosewith hundreds of digits.Carry ‑ Structured Block Factorization establishes a new category of structural factorization,distinct from classical number ‑ theoretic algorithms, and offers a theoreticalfoundation for understanding how integers encode their multiplicative origins.