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 V₁₄ₓ₀, ₊ (m) = U (m) - beta^-1sK (m), where sK (m) is the block entropy density. We formulate the corresponding free-energy crossing problem, derive the renormalized central-band gap a₁₄ₓ₀, ₃₄₋ₓ₀, ₊, 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.
S. Pan (Wed,) studied this question.
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