We study phase crossing in a two-dimensional interacting block-spin strip with left boundary fixed in the negative phase and right boundary fixed in the positive phase. The model consists of K-spin blocks on a rectangular lattice, with a double-well onsite cost for each block magnetization and quadratic nearest-neighbor coupling in both coordinate directions. We prove that the minimal strip-crossing energy scales as Lᵧ*K*sqrt (J), where Lᵧ is the transverse strip width, and that the total central-band volume of any minimizer scales as Lᵧ*sqrt (J). Thus the one-dimensional mesoscopic crossing layer lifts to a two-dimensional mesoscopic crossing wall. The result should be read as the first static step toward a higher-dimensional path-geometry program rather than as a complete theory of two-dimensional metastable dynamics. Related earlier works by S. Pan are available at DOI: 10. 5281/zenodo. 19673404, DOI: 10. 5281/zenodo. 19689210, and DOI: 10. 5281/zenodo. 19690441.
S. Pan (2026) studied this question.