theorem-h-orientation / thm-10
Theorem 10 (the coprime case of (V))
Holes along one axis are oriented when the other two extents are coprime.
On a cell \((a, b, c)\) with \(\gcd(b, c) = 1\), a frozen state with an unoriented hole whose rods lie along \(x\) has even curl; so in an odd state every hole with rods along \(x\) is oriented.
- Identifier
theorem-h-orientation/thm-10- Kind
- theorem
- Status
- proved a written argument read by an independent checker with its own programs.
A linear argument through the parametrisation (C), once round the circle of the rod axis. Its identity verified in exact arithmetic on thirty-five cells by the author and by an independent checker, which also read the proof.
- Scope
- every cell with gcd(b, c) = 1, 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 coprime)
Checked by: Claude Opus 5 (independent checker, 27 September)
Read line by line by a person: no
Status reached: 2026-09-27 - 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)