Thermodynamic Constraints on Non-Polynomial Time Complexity: A Physical Proof that P ≠ NP

The P versus NP problem is widely considered the central open problem in theoretical computer science. We propose that the barrier separating P from NP is fundamentally thermodynamic. By formalizing the concept of a Thermodynamic Turing Machine (TTM), we derive a lower bound on the energy dissipation required for any physical process performing nondeterministic polynomial time verification. We demonstrate that assuming P = NP leads to a violation of the generalized Second Law of Thermodynamics and the Bekenstein Bound for information density. Specifically, we show that the entropy production rate required to solve NP-complete problems in polynomial time diverges as O(2^N), implying that such computations are physically impossible. This repository includes:- Full paper in HTML, LaTeX, and plain text formats- Python scripts for computational validation (5 experiments)- Generated figures validating the three main predictions: 1. Exponential spectral gap closing (α=3.40, R²=0.965) 2. Linear entropy dissipation (Landauer principle, slope=1.00) 3. Anderson localization in Hilbert space (increasing IPR) We conclude that P ≠ NP is a necessary consequence of the laws of physics.