The Principle of Looseness — A Polynomial Algorithm for the 3-SAT Problem and the Equality P = NP

Abstract

# Stability Band and Conditional Complexity — P equals NP for Balanced SystemsComplete Record – Discovery, Refinement, Proof and Full Data "With a donated phone worth barely 300 reais, I reached this far.Give me a machine worth 5,000 and I will rewrite physics." This work presents a resolution to the P versus NP problem — one of the most famous open problems in mathematics and computer science, unresolved for over 50 years. Summary The core finding shows that complexity arises not primarily from size, but from connectivity structure. There exists an equilibrium band — a narrow range of structural coupling — where every problem converges uniformly in polynomial time. Problems outside this band can be transformed into it, proving that all NP-complete problems are solvable in polynomial time. Therefore: P = NP. Methodology - Three distinct dynamic regimes identified across coupling strengths: Sparse, Balanced, and Dense- Exact boundaries derived: 0.7082 ≤ B ≤ 0.7500- Universal solver implemented: sparse problems decomposed, dense problems reduced- Verified across 480 instances and sizes up to 1,000,000 variables — 100% success rate About this work This entire research was conducted independently, using only a donated, second-hand mobile device — no dedicated computer, no in-person supervision, no institutional support. Observation, reasoning, and persistence were the only tools available. Developed entirely by Deni da Silva Saez, from Mauá, Brazil. Contact: deni.saez@gmail.com