Structural Constraints on Gain and Convergence in Greedy Max-Cut

This paper develops a structural framework for analyzing the convergence of the randomized greedy vertex-flip algorithm for the Max-Cut problem, and forms part of an ongoing research program on the structural and spectral analysis of greedy algorithms in combinatorial optimization. A central obstacle in prior analyses is the reliance on lower bounds for total positive gain, which are not known to hold uniformly across all configurations. This work addresses that issue by introducing a refined structural gain dichotomy that incorporates the distribution of local fields, capturing how alignment is spread across vertices. The analysis shows that configurations cannot simultaneously exhibit small gain, limited misalignment, and high energy. This yields a three-way structural decomposition in which every configuration must either exhibit substantial gain, widespread misalignment, or lie in a low-energy regime. A key contribution is demonstrating that misalignment can be converted into expected improvement, providing a unified structural mechanism that prevents stagnation away from high-quality configurations. This leads to convergence guarantees via a monotone energy argument, without requiring explicit lower bounds on total gain. These results strengthen the connection between local alignment, spectral structure, and global energy, and represent progress toward an unconditional structural theory of greedy local search. This work extends a prior preprint introducing a structural gain dichotomy for greedy Max-Cut, and develops a refined framework incorporating alignment-based constraints and misalignment-driven improvement.