This paper presents a formal analysis of the P versus NP problem through the integration of computational complexity, statistical mechanics, and stochastic thermodynamics. By mapping the solution space of the random 3-SAT problem at the clustering threshold (αd ≈ 3. 921) to a Directed Acyclic Graph, the minimum memory requirement is evaluated using the Reversible Pebbling Game framework. The analysis indicates that restricting an algorithm to polynomial space necessitates an exponential number of irreversible erasure operations to traverse the fractured energy landscape. Under the Physical Church-Turing Thesis and Landauer's Principle, this erasure translates to an exponentially scaling thermodynamic dissipation, which violates the physical constraints of polynomial-time computation. Furthermore, a corollary demonstrates that in a purely abstract (Platonic) Turing machine model, these forced erasures dictate an exponential lower bound on discrete computational time steps. This repository also includes a supplementary Python script that provides an interactive empirical visualization of the theoretical crossover points (N₂ₑ₈ₓ) discussed in the manuscript.
J. Lemos (Mon,) studied this question.