Universal Crossover Scaling of the P–NP Transition and the Computational Singularity: A Complete Empirical and Theoretical Synthesis

The P versus NP problem remains one of the most profound openquestions in mathematics and theoretical computer science. While formal proofs have been elusive, a growing body of theoretical and experimental work indicates that the exponential difficulty of NP-completeproblems originates from a universal geometric and topological obstruction in their solution spaces. This paper presents the definitive synthesis of a comprehensive experimental campaign, establishing the most complete empirical foundation to date for the conjectureP ̸= NP.We report the discovery of a universal crossover scaling law governing the continuous transition from 2-SAT (in P) to 3-SAT (NPcomplete): pc(n) = A·n−δeff(n), with A = 2.70±0.15, δ∞ = 1.38±0.05,δ0 = 2.94 ± 0.10, nc = 2.77 ± 0.20 (R2log = 0.956 on 18 independentpoints). This law demonstrates that in the thermodynamic limit, anarbitrarily small admixture of 3-SAT clauses suffices to fragment thesolution space.We provide the first direct experimental detection of persistent second Betti numbers in 3-SAT: β2 = 438 persistent two-dimensionalholes at n = 100. Multi-metric quantification of the Overlap GapProperty (OGR = 4.62 ± 1.46 on SATLIB benchmarks, Marchenko–Pastur heavy tails, and statistical tests with p < 10−150) confirms1strong fragmentation. Replica exchange Monte Carlo reveals reducedswap acceptance rates and elevated energy autocorrelation, characteristic of Replica Symmetry Breaking (RSB). A geometric taxonomy ofNP-complete problems demonstrates that only the k-SAT family resides in the strong RSB phase.We establish a precise mathematical correspondence between theobserved computational phase transition and the laws governing blackhole singularities: the crossover law mirrors Choptuik’s critical scaling, the persistent β2 holes mirror Bekenstein–Hawking entropy, andthe OGP threshold mirrors Penrose’s trapped surface condition. Thiscorrespondence suggests that the P vs NP barrier is a physical law ofthe computational universe.We propose that the only conceivable resolution of the P vs NPproblem—should one exist—would require accessing an “extra dimension” capable of resolving the topological obstructions, effectively bringing the computational singularity to a point. The impossibility of sucha reduction in polynomial time is precisely the statement that P ̸= NP.This perspective reframes the problem as an observable physical phasetransition and provides a concrete roadmap for future rigorous mathematical proofs.The complete dataset supporting this study is provided in the accompanying Zenodo repository (DOI: 10.5281/zenodo.19687142). Code used for data generation and analysis is available upon reasonable request.