Planar Crossing Walls in a Two-Dimensional Interacting Block-Spin Strip

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 L_y*K*sqrt(J), where L_y is the transverse strip width, and that the total central-band volume of any minimizer scales as L_y*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.