Formal Specification, Correctness Argument, and Complexity Analysis of Recursive SAT Decomposition

This manuscript presents a proposed recursive decomposition framework for Boolean satisfiability (SAT), together with a formal specification, correctness argument, and complexity analysis. The framework recursively partitions a Boolean satisfiability instance into smaller child instances, uses a finite base-case library, and combines child information through parent-level Boolean placeholders and shared-variable compatibility checks. The manuscript develops the intended SAT-completeness and UNSAT-soundness arguments, including the preservation of variable identity and joint compatibility across decomposition layers. It also provides an operation-count analysis and discusses the conditions required to establish a polynomial-time implementation. This work is submitted as a proposed mathematical argument for independent review. In particular, the decomposition equivalence, completeness of the compatibility procedure, and polynomial bit-complexity warrant rigorous examination. The manuscript does not claim that the P versus NP problem has been accepted as solved.