P is Different from NP

This paper presents a self-contained proof that P is different from NP, circumventing the established barriers of relativization, natural proofs, and algebrization. We study the computational complexity of satisfiability through a structural analysis of clause interactions in conjunctive normal form (CNF) formulas. We introduce a geometric and algebraic framework in which clauses are represented as sets over the space of primary clauses, making clause intersections explicit and enabling an exact formulation of satisfiability via the inclusion–exclusion principle. Within this framework, we define several families of CNF languages characterized by their numerical capacity and by the degree of clause intersections. We show that these structural parameters induce a strict hierarchy of satisfiability problems. For the language V, defined by formulas of exact numerical capacity, satisfiability can be decided in polynomial time, while linear-time decision is provably insufficient. For the language A, obtained by extending formulas in V with an additional clause, we show that bounding the intersection degree yields polynomial-time decidability, whereas in the unrestricted case no polynomial-time deterministic Turing machine can decide satisfiability. Our results establish explicit connections between clause-intersection structure, algebraic representations of SAT, and fundamental limits on step-wise information acquisition by deterministic machines.