theorem-h-orientation / thm-11
Theorem 11 (greatest common divisor two, odd length)
Holes along one axis are oriented when the other two extents have greatest common divisor 2 and the axis has odd length.
On a cell \((a, b, c)\) with \(\gcd(b, c) = 2\) and \(a\) odd, a frozen state with an unoriented hole whose rods lie along \(x\) has even curl.
- Identifier
theorem-h-orientation/thm-11- Kind
- theorem
- Status
- proved a written argument read by an independent checker with its own programs.
Needs integrality through a parity count on the halves of a plane; the parity of \(a\) enters. Its identity verified in exact arithmetic on thirty-three cells by two implementations; read by an independent checker.
- Scope
- gcd(b, c) = 2 and a odd, for holes with rods along x
Hypotheses: frozen - 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
- no claim in a published manifest
- Artefacts
- Provenance
- Written by: Claude Opus 5 (agent gtwo)
Checked by: Claude Opus 5 (independent checker, 27 September)
Read line by line by a person: no
Status reached: 2026-09-27 - Notes
With \(a\) even the vertex \(P\) of the proof does not exist and the real relaxation has solutions; integrality is needed and no replacement for the parity count is known. Theorems 10 and 11 give no case of Theorem H that needed them.
- 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)