Thermodynamic Erasure Complexity in Spin Glass Landscapes: A Reversible Pebbling Analysis of the P versus NP Problem

This preprint proposes a physico-computational framework evaluating the P versus NP question for physical computers 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 (alpha_d ≈ 3.921) to a Directed Acyclic Graph, the minimum memory requirement is evaluated using the Reversible Pebbling Game framework. The analysis indicates that restricting a classical deterministic 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, suggesting a physical constraint on polynomial-time execution. Furthermore, a corollary outlines how in an abstract (Platonic) Turing machine model, these forced erasures dictate an exponential lower bound on discrete computational time steps. Updates in Version 2.0.0 (September 2026):1. Machine-Checked Formalization (Lean 4): The combinatorial core of the Reversible Pebbling lower bound (`landauer_erasure_lower_bound`) has been formalized and verified without unproven tactics (`sorry`) using the Lean 4 theorem prover (v4.33.1).2. Clarified Scope: Distinguishes between physical constraints on deterministic hardware and abstract computational models.3. Supplementary Materials: Includes full Lean 4 source files (`PebblingGame.lean`), verification scripts, and empirical Python crossover ($N_{crit}$) visualization tools in the supplementary package (`Lemos_P_vs_NP_Supplementary_Code.zip`).