We isolate the final analytical transmission step in the wall-rigidity program for a two-dimensional interacting block-spin strip. Previous work established the static wall law and then developed a wall-rigidity preprint in which the main geometric mechanism was made explicit and the remaining finite-J closure was reduced to a spectral problem after translation fixing. The present note records that spectral closure machinery in a separate technical form. We formulate wall-scale compactness of the reference layers, local convergence of the discrete curvatures, Riemann-sum convergence of the discrete bilinear form, passage of the translation orthogonality constraint, and the resulting conditional spectral contradiction theorem. The note should be read as a technical companion to the two-dimensional wall-rigidity preprint rather than as a separate geometric 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, and DOI: 10.5281/zenodo.19696148.
S. Pan (Wed,) studied this question.
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