Variance Mediation and Geometric Susceptibility Near the P–NP Transition: An Experimental Study

We suggest that the intermediate variance triangle represents a promising experimental frontier for complexity theory, offering a geometric and statistical lens on the P→NP transition that complements—but does not replace—classical reduction-based approaches. This work presents a sequence of controlled computational experiments probing the behavior of Boolean satisfiability problems along a continuous interpolation from 2-SAT (polynomial-time) to 3-SAT (NP-complete). Rather than treating the P→NP transition as a discrete boundary, we study it as a geometric phase transition characterized by changes in instance-level variability, susceptibility, and solver response. Across multiple experimental layers—including inverse variance tracking, perturbative stability analysis, and solver memory freezing—we demonstrate that variance acts as an active mediating quantity in the transition region. While mean computational effort increases gradually, variance and susceptibility exhibit sharp amplification prior to visible hardness escalation. This anticipatory behavior mirrors critical phenomena in physical phase transitions. Crucially, we show that variance amplification persists across solver architectures and becomes dramatically stronger when solver memory is removed, indicating that difficulty structure is intrinsic to instance geometry rather than an artifact of heuristics or learning mechanisms. Solver memory systems moderate and mask this geometric variance but do not create it. These results support an interpretation of NP-hardness as a regime of structural instability and corridor fragmentation in solution space, where small perturbations induce large performance fluctuations. The findings do not resolve the P vs NP problem, but they provide empirical evidence that any separation—if present—manifests through loss of geometric stability rather than abrupt changes in average-case complexity. Related work: De Jesus, Elias. (2026). Encoding-Invariant Mode Geometry in Constraint Satisfaction Problems. Zenodo. https://doi.org/10.5281/zenodo.18261772