Entropic Renormalization of Crossing Layers in a One-Dimensional Block-Spin Chain

We study finite-temperature corrections to the crossing-layer theory of a one-dimensional interacting block-spin chain. Previous static work shows that the minimal phase-crossing energy scales as K*sqrt(J) and that any optimal crossing layer has width of order sqrt(J). The present paper identifies the corresponding entropic correction at the block level. We show that the dominant entropy of a mesoscopic crossing layer is not of order sqrt(J)*log(K), but of order K*sqrt(J), because each block magnetization carries an internal binomial multiplicity. This leads naturally to an effective free-energy density Vbeta,K(m) = U(m) - beta⁻¹s_K(m), where s_K(m) is the block entropy density. We formulate the corresponding free-energy crossing problem, derive the renormalized central-band gap abeta,delta,K, and obtain the associated free-energy tradeoff and mesoscopic layer scale. The paper should be read as a finite-temperature renormalization of the static crossing-layer theory rather than a complete dynamical slowdown theorem. Related earlier works by S. Pan are available at DOI: 10.5281/zenodo.19673404 and DOI: 10.5281/zenodo.19689210.