theorem-h-orientation / parametrisation-c
The parametrisation (C) (completeness of the chain modes)
Every real solution of the type equations is a weighted sum of chain modes plus two constants.
Every real solution of the one-hot equations and (T2) on a cell is \(e_x\) plus a sum of chain modes (a weight on every chain of every axis) plus two antisymmetric constants: \(Y = F_x - F_z + \sigma\alpha\), \(Z = F_y - F_x + \sigma\beta\).
- Identifier
theorem-h-orientation/parametrisation-c- Kind
- theorem
- Status
- proved a written argument read by an independent checker with its own programs.
Proved for every cell by a stratified symbolic kernel computation on the unit torus (68 strata) and confirmed by exact ranks on cells (17 September); on 2026-10-01 an independent checker (agent p1check) reproduced the proof with its own model and its own stratification and exact ranks on twelve cells, closing one expository gap in the companion kernel result.
- Scope
- every cell
- Depends on
- The model (first section; Holes, cages and rods) proved theorem-h-orientation/model
- Used by
- Statement R, a monodromy in the weights, and the heights are not a route to (V) (Theorem U section, item 9) closed-route theorem-h-orientation/closed-r-then-monodromy
- Theorem 1 (the reduction), form of 30 September 2026, with Lemmas 22 to 25 proved theorem-h-orientation/thm-1
- 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 15 proved theorem-h-orientation/thm-15
- Theorem 22, with Theorems 20 and 21 (cells with two chains in every pair of planes) proved theorem-h-orientation/thm-22
- The weights of (C) are rarely integral (correction of 30 September 2026) corrected theorem-h-orientation/weights-integrality-correction
- Artefacts
- Provenance
- Written by: Claude Opus 5 (agent completeness)
Checked by: Claude Opus 5 (agent p1check)
Read line by line by a person: no
Status reached: 2026-10-01 - 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)