Rigidity of Crossing Walls in a Two-Dimensional Interacting Block-Spin Strip

We study rigidity of near-optimal crossing walls in a two-dimensional interacting block-spin strip. A previous static wall theorem established that the minimal strip-crossing energy is of order L_y*K*sqrt(J) and that every minimizer contains a mesoscopic wall of total central-band volume of order L_y*sqrt(J). The present paper addresses the next question: whether such walls must be approximately planar. We prove coarse normalization for good rows and a rigidity theorem in the normalized translated-wall model, and we reduce the full wall-rigidity statement to a spectral closure after translation fixing. The paper should be read as a two-dimensional wall-rigidity preprint whose main geometric mechanism is established explicitly, while the final finite-J spectral closure is formulated and pushed to a discrete-to-continuum contradiction scheme in the technical sections and appendices. Related earlier works by S. Pan are available at DOI: 10.5281/zenodo.19673404, DOI: 10.5281/zenodo.19689210, DOI: 10.5281/zenodo.19690441, and DOI: 10.5281/zenodo.19693231.