Quantitative Structural Constraints and Second-Moment Forcing in Greedy Max-Cut

This paper studies the convergence behavior of the randomized greedy vertex-flip algorithm for the Max-Cut problem through a structural and spectral analysis. A central challenge in analyzing greedy Max-Cut is the lack of uniform lower bounds on total positive gain, which governs the rate of improvement. Existing approaches often rely on such bounds as assumptions, leaving a gap in the theoretical understanding of convergence. In this work, we develop a quantitative structural framework relating gain, imbalance, and energy through local alignment variables. We establish concentration bounds showing that alignment cannot concentrate near neutrality when imbalance is present, and derive a spectral lower bound on the second moment of the alignment variables. Combining these results, we introduce a second-moment forcing mechanism: dispersion in local fields forces either substantial total gain or significant expected improvement. This provides a structural explanation for why greedy Max-Cut cannot stagnate away from near-stationary configurations and yields convergence guarantees without requiring explicit lower bounds on gain. This work is part of an ongoing research program developing a quantitative structural theory of greedy local search in combinatorial optimization. Future work will aim to derive sharper bounds on expected improvement, including potential quadratic relationships between improvement and imbalance.