Mesoscopic Wall Shells Are Dynamically Unavoidable in a Two-Dimensional Interacting Block-Spin Strip

We study dynamical crossing paths in a two-dimensional interacting block-spin strip under local single-spin-flip dynamics. Previous work established that the minimal strip-crossing energy is of order Lᵧ*K*sqrt (J), that every minimizer contains a mesoscopic wall of total central-band volume Lᵧ*sqrt (J), and that near-optimal walls are rigid up to a controlled number of kinks. The present paper addresses the corresponding dynamical question: whether every near-optimal minus-to-plus switching path must traverse a mesoscopic wall shell. We prove that any path whose maximal energy remains within A*K*sqrt (J) of the optimal strip-crossing barrier must contain an intermediate configuration whose rowwise central-band structure forms a wall shell of mesoscopic width and controlled rigidity. Thus the static wall is not merely an optimizer of a boundary-value problem: it is dynamically unavoidable along low-barrier switching paths. The paper should be read as a two-dimensional dynamical bridge preprint rather than as a full bottleneck theorem. Related earlier works by S. Pan are available at DOI: 10. 5281/zenodo. 19673404, DOI: 10. 5281/zenodo. 19689210, DOI: 10. 5281/zenodo. 19690441, DOI: 10. 5281/zenodo. 19693231, DOI: 10. 5281/zenodo. 19696148, and DOI: 10. 5281/zenodo. 19697539.