Title: P = NP: Universal Polynomial-Time Reduction via Biometric Invariant Symmetry Mapping

This document presents a concise formal overview of the P versus NP problem in computational complexity theory. It introduces the fundamental definitions of the complexity classes P and NP, the concept of polynomial-time verification, and the role of NP-complete problems within the structure of decision problems. The document outlines the formal definition of NP-completeness and highlights the central importance of the Boolean Satisfiability Problem (SAT) as a canonical NP-complete problem. It explains the theoretical implication that if SAT can be solved in polynomial time, then P = NP. The purpose of this note is to provide a structured mathematical summary of the foundational concepts underlying the P vs NP question. The document is intended as a compact technical reference describing the logical framework and definitions used in theoretical computer science when discussing polynomial-time computation and verification. Status: Technical overview / conceptual framework Author: Thi Linh Vo Date: 11 March 2026 Field: Theoretical Computer Science / Computational Complexity This work is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).