theorem-h-orientation / cage-laws
Law C, law \(P^k\), Lemmas 1 to 3 (the cage laws)
The local linear laws the frozen condition imposes on the types round each hole, with the 1,200 possible patterns of one cage.
The linear identities between the type indicators of the vertices of a cage that (T2) implies, including the law \(P^k\) relating the rods and bridgeheads of a hole, and Lemma 3, the enumeration of the 1,200 patterns of one cage.
- Identifier
theorem-h-orientation/cage-laws- Kind
- lemma
- Status
- proved a written argument read by an independent checker with its own programs.
Written arguments and a finite enumeration run by at least two independent programs.
- Scope
- every cell
- Depends on
- The model (first section; Holes, cages and rods) proved theorem-h-orientation/model
- Used by
- No weighted sum of laws proves (V) at \(g \geq 2\) (Theorem U section, items 3 to 5) closed-route theorem-h-orientation/closed-linear-relaxation
- The kink counts per chain class: the wave equation and one defect per axis (Theorem U section, item 10) proved-unchecked theorem-h-orientation/kink-counts-per-class
- 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 23, with Lemmas 13 to 20 and Theorems 16 to 19 (the signed moments) proved theorem-h-orientation/thm-23
- Theorems 4 to 8 (line counts, two separable functions with product zero, walls, uniform chains beside a wall) proved theorem-h-orientation/thm-6-separability
- Theorem 9 (the certificate of the diagonal law, closed form) proved theorem-h-orientation/thm-9-diagonal-law
- Provenance
- Written by: Claude Opus 5 (agents of 27 September)
Checked by: Claude Opus 5 (agents check, check2)
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)