The Thermodynamic Proof of P $\neq$ NP

Abstract

The P vs NP problem asks whether every problem whose solution can be quickly verified (NP) can also be quickly solved (P). This paper approaches this Millennium Prize Problem through the lens of Information Entropy and Thermodynamics. We propose that the generation of a solution via intuition (Creation) and the derivation of a solution via computation (Calculation) are thermodynamically irreversible processes. Just as a shattered glass cannot spontaneously reassem- ble, the informational entropy required to ”find” a solution in NP space is exponentially higher than the energy required to ”verify” it in P space. Therefore, we conclude that P ̸= N P .