theorem-h-orientation / thm-m-prime

Theorem M′ (from the project’s earlier record)

If one sublattice lacks a type that the other has, the state has even curl.

Identifier

theorem-h-orientation/thm-m-prime

Kind

theorem

Status

proved a written argument read by an independent checker with its own programs.

Two proofs, with lemmas on four hexagons that carry DRAT certificates, and one second reader; confirmed on every cell by an audit of 27 September by both proofs.

Scope

every cell
Hypotheses: frozen

Depends on

• The model (first section; Holes, cages and rods) proved theorem-h-orientation/model

Used by

• Theorem 1 (the reduction), form of 30 September 2026, with Lemmas 22 to 25 proved theorem-h-orientation/thm-1
• Theorem 1, first form (27 September 2026), through K_u proved theorem-h-orientation/thm-1-first-form
• Theorem 10 (the coprime case of (V)) proved theorem-h-orientation/thm-10
• Theorem 11 (greatest common divisor two, odd length) proved theorem-h-orientation/thm-11
• 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 22, with Theorems 20 and 21 (cells with two chains in every pair of planes) proved theorem-h-orientation/thm-22
• Theorem 23, with Lemmas 13 to 20 and Theorems 16 to 19 (the signed moments) proved theorem-h-orientation/thm-23

Provenance

Written by: Claude (Fable and Opus 5), September 2026
Checked by: one second reader (agent); audit of 27 September (agent noncubic and a second reader)
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)