Hexagonal Stacking Maximizes Proton Configurational Entropy among Ice-I Polytypes

Abstract

Ice I admits cubic, hexagonal, and mixed layer stackings, but rigorous entropy comparisons have focused on the two ideal endmembers. We represent every cyclic uniform-registry stacking by a word in a nonnegative transfer operator K and its transpose. For every such even-length word, applying the Schatten-Hölder inequality proves that alternating hexagonal stacking maximizes the ice-rule count at every common finite cross-section; the configuration constant is therefore maximal among all periodic uniform-registry polytypes. We obtain the lower endpoint by restricting Nagle's positive even-subgraph expansion to exactly enumerated disjoint blocks. Finner's degree-two hypergraph Hölder inequality and rational Collatz-Wielandt certificates for two-replica prism transfer operators give the upper endpoints. These constructions yield $1.503360 \le w \le 1.540196$, with $w(\mathrm{Ic}) \le 1.527699$.