Optimal Crossing Layers in a One-Dimensional Interacting Block-Spin Chain

We study the minimal energy required for a phase-crossing block profile in a one-dimensional interacting block-spin chain. The model consists of L blocks of K Ising spins, with a double-well onsite cost for each block magnetization and a nearest-neighbor quadratic coupling penalizing sharp spatial variation. We consider the variational problem of connecting the negative and positive phases under fixed boundary conditions and prove that, in the mesoscopic regime 1 <= J <= K^2, the optimal crossing energy scales as K*sqrt(J), while the optimal transition-layer width scales as sqrt(J). The lower bound follows from an energetic tradeoff between central-band occupancy and profile roughness; the upper bound is obtained by an explicit staircase trial profile. We also record the formal continuum scaling linking the discrete crossing problem to a one-dimensional interface functional. More broadly, the paper provides a static crossing theory for an interacting one-dimensional block-spin chain and prepares the spatial layer mechanism needed for later dynamical work on metastable switching. A previous zero-dimensional reversible prototype by S. Pan is available at DOI: 10.5281/zenodo.19673404.