A Proof of P ≠ NP via Self-Reference Incompressibility and Information-Theoretic Lower Bounds.pdf

We present a complete proof that P ̸= NP by establishing a fundamental information-theoretic barrier via proof by contradiction. Main argument: Assume P = NP. Then polynomial-time algorithms have bounded output complexity (O(log n) bits beyond the instance via trace encoding), while NP-complete problem witnesses require ω(log n) bits of conditional information (for TSP: Θ(n log n) bits via anti-concentration and counting arguments). This contradiction establishes P ̸= NP. The proof rests on three pillars: 1. Self-Reference Incompressibility (S.R.I.): Any polynomial-time algo- rithm producing output y from input x satisfies K(y | x) ≤ O(log n) via trace encoding, where K(· | ·) denotes conditional Kolmogorov complexity. 2. TSP Conditional Incompressibility: For Euclidean TSP, optimal tours sat- isfy K(π ∗ | x) = Θ(n log n) with high probability, proven via anti-concentration (Milnor-Thom bounds) and incompressibility counting arguments. This holds even given the instance x. 3. Universal Inheritance Theorem (U.I.T.): Conditional incompressibility propagates through Karp reductions: all NP-complete problems inherit ω(log n) witness complexity bounds.