The Synthesis of Linear Algebra and Hierarchical Parity in Solving the P vs. NP Question

The $P$ vs. $NP$ question remains the most significant unsolved problem in theoretical computer science. This paper introduces the Hierarchical Parity-Corrected Matrix Inversion (HPMI) method. By transforming Boolean satisfiability into a structured binary matrix system $A\mathbf{x} = \mathbf{b} \pmod 2$, and applying a 1-3-9 geometric scaling constraint, we demonstrate that the “exponential wall” is a product of search-based algorithms rather than the inherent nature of the problems. We provide evidence that $NP$ problems can be resolved in $O(n^\omega)$ time, effectively proving $P = NP$.