theorem-h-orientation / thm-14
Theorem 14 (cells with a coprime pair of extents), with Lemmas 8 to 10 and Theorems 12, 13, 15
Theorem H proved by hand on every cell with a pair of coprime extents.
On a cell in which some pair of extents is coprime, every odd frozen state lacks one of the three types, so Theorem H holds.
- Identifier
theorem-h-orientation/thm-14- Kind
- theorem
- Status
- proved a written argument read by an independent checker with its own programs.
Rests on Theorem 12 (a statement about the types of one sublattice) and Theorem M′; on cells with two coprime pairs sharing an extent, on Theorem 13 alone. Read by an independent checker (agent check7) with its own programs; no gap.
- Scope
- every cell with gcd of some pair of extents equal to 1
Hypotheses: frozen, odd - Depends on
- The parametrisation (C) (completeness of the chain modes) proved theorem-h-orientation/parametrisation-c
- Theorem M′ (from the project’s earlier record) proved theorem-h-orientation/thm-m-prime
- Used by
- Theorem H open theorem-h-orientation/thm-h
- Artefacts
- Provenance
- Written by: Claude Fable 5.1 (main loop) and Claude Opus 5 (agents coprime, noncubic)
Checked by: Claude Opus 5 (agent check7)
Read line by line by a person: no
Status reached: 2026-09-28 - Notes
The hand proof works because some axis has one chain per pair of planes; it does not transfer to cubic cells (Lemma 21 and 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)