theorem-h-orientation / thm-h
Theorem H
A frozen state with an odd hexagon has an axis that is either unused or used with one sign only.
In a frozen state with a hexagon of odd curl, some axis is empty (no vertex has that type) or one-way (every vertex of that type has the same sign of moment).
- Identifier
theorem-h-orientation/thm-h- Kind
- theorem
- Status
- open a target with no verdict.
Proved on the cells with a coprime pair of extents (Theorem 14), on the cells with two chains in every pair of planes (Theorem 23), and by certificate on the cubic cells of side one to eight and on four cells that are not cubic with their divisors (Theorems 2 and 24). Open elsewhere, where it follows from Theorem U by Theorem 1.
- Scope
- every cell; open on the cells with no coprime pair and more than two chains in every pair of planes that are not covered by a certificate, the cubic cells of side nine and more among them
Hypotheses: frozen, odd - Depends on
- Theorem 1 (the reduction), form of 30 September 2026, with Lemmas 22 to 25 proved theorem-h-orientation/thm-1
- Theorem 14 (cells with a coprime pair of extents), with Lemmas 8 to 10 and Theorems 12, 13, 15 proved theorem-h-orientation/thm-14
- Theorem 23, with Lemmas 13 to 20 and Theorems 16 to 19 (the signed moments) proved theorem-h-orientation/thm-23
- Theorem 2 certified theorem-h-orientation/thm-2
- Theorem 24 certified theorem-h-orientation/thm-24
- Used by
- Periodicity of frozen zero-flux states (first section, with Theorem P and Lemma A of the earlier note) proved theorem-h-orientation/periodicity
- Provenance
- Written by: Lyndon Drake (statement)
Checked by: no independent checker
Read line by line by a person: no
Status reached: 2026-10-01 - Notes
The target statement of the note. Its standing by kind of cell is the table in the note’s first section.
- Source
- Theorem H for frozen pyrochlore ice: proofs on the cells with a coprime pair of extents and on the cells with two chains in every pair of planes, a reduction to Theorem U alone on every cell, and the cubic cells of side one to eight, version 2026-10-01; manifest (JSON)