Heat transport in a hierarchy of reduced-order convection models

The paper rigorously defines the HK hierarchy, proves structural facts (e.g., equivalence of two Nusselt-number formulas in the distinguished setting and the vanishing of shear modes on primary equilibria), and provides strong computational evidence via SOS bounds that primary equilibria maximize heat transport near onset. It also gives the exact Lorenz-like formula NLmn = 3 − 2Rmn/R. However, it does not prove a uniform δ > 0 for all HKMi guaranteeing that N* is attained by primary equilibria for Rc ≤ R < Rc(1+δ). The model solution outlines a plausible center-manifold/normal-form argument to establish such a δ and the maximizing property, but key steps (global attractor contained in the center-manifold neighborhood; sign of the cubic coefficient uniformly across the hierarchy) are assumed rather than proved. Hence both the paper and the model solution fall short of a complete proof of the stated optimization claim.