Hexagonal Stacking Maximizes Proton Configurational Entropy among Ice-I Polytypes
Abstract
Ice I admits cubic, hexagonal, and mixed layer stackings, but rigorous comparisons of its proton configurational entropy have focused on the two ideal endmembers. We identify every cyclic uniform-registry ice-I stacking with a word in a nonnegative layer-transfer operator and its transpose. Applying the Schatten-Hölder inequality to every even-length word proves that alternating hexagonal stacking maximizes the ice-rule configuration count at each common finite cross-section. Consequently, hexagonal ice maximizes the thermodynamic configuration constant among all periodic uniform-registry ice-I polytypes. We also construct rigorous lower and upper bounds without finite-size extrapolation. Restricting Nagle's positive even-subgraph expansion to disjoint blocks gives the lower bound; Finner's hypergraph Hölder inequality and rational Collatz-Wielandt certificates for two-replica prism operators give the upper bounds. The resulting configuration constant w, whose logarithm is the entropy per molecule in units of Boltzmann's constant, satisfies 1.503360 ≤ w ≤ 1.540196, with the sharper cubic-ice ceiling w ≤ 1.527699. The same bracket controls lower and upper configuration growth rates of fixed aperiodic stackings along the specified prism-tileable exhaustions. These results establish hexagonal stacking as a possibly nonunique maximizer; equality of the cubic and hexagonal entropy constants remains unresolved.
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BibTeX
@misc{chen2026hexagonal,
title = {{Hexagonal Stacking Maximizes Proton Configurational Entropy among Ice-I Polytypes}},
author = {Zhengyue Chen and Sheng Ran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.07613},
eprint = {2608.07613},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.07613},
}
RIS
TY - UNPB
TI - Hexagonal Stacking Maximizes Proton Configurational Entropy among Ice-I Polytypes
AU - Zhengyue Chen
AU - Sheng Ran
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2608.07613
UR - https://arxiv.org/abs/2608.07613
AB - Ice I admits cubic, hexagonal, and mixed layer stackings, but rigorous comparisons of its proton configurational entropy have focused on the two ideal endmembers. We identify every cyclic uniform-registry ice-I stacking with a word in a nonnegative layer-transfer operator and its transpose. Applying the Schatten-Hölder inequality to every even-length word proves that alternating hexagonal stacking maximizes the ice-rule configuration count at each common finite cross-section. Consequently, hexagonal ice maximizes the thermodynamic configuration constant among all periodic uniform-registry ice-I polytypes. We also construct rigorous lower and upper bounds without finite-size extrapolation. Restricting Nagle's positive even-subgraph expansion to disjoint blocks gives the lower bound; Finner's hypergraph Hölder inequality and rational Collatz-Wielandt certificates for two-replica prism operators give the upper bounds. The resulting configuration constant w, whose logarithm is the entropy per molecule in units of Boltzmann's constant, satisfies 1.503360 ≤ w ≤ 1.540196, with the sharper cubic-ice ceiling w ≤ 1.527699. The same bracket controls lower and upper configuration growth rates of fixed aperiodic stackings along the specified prism-tileable exhaustions. These results establish hexagonal stacking as a possibly nonunique maximizer; equality of the cubic and hexagonal entropy constants remains unresolved.
ER -