The model (first section; Holes, cages and rods)

Definitions: cells, ice states, vertex types, hexagons and their curl, frozen and odd states, holes and their rods.

A cell \((a, b, c)\) of pyrochlore ice is the torus of quarter-cube coordinates modulo \((4a, 4b, 4c)\); its vertices are the sublattice A (even coordinates with sum \(0 \bmod 4\)) and B \(=\) A \(+ (1,1,1)\); each vertex carries four bonds in the directions \(d_0, \dots, d_3\); an ice state gives every bond a direction with two in and two out at every vertex; the dipole moment of a vertex is the sum of its outgoing bond directions and its type is the axis of that moment; a hexagon is the closed walk of six bonds round an octahedral hole determined by two of the four directions; a state is frozen when no hexagon circulates, and (T2) is the matching condition that expresses this in the types; a hexagon’s curl is odd or even; an odd state has an odd hexagon; the rods of a hole are the two vertices on its axis with the hole’s axis as type, its bridgeheads the four vertices beside them, and a hole is oriented when one rod points towards it and one away.

Identifier

theorem-h-orientation/model

Kind

definition

Status

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

Definitions. Checked for consistency with the simulation code by several independent models, most recently h730_lib (agent pacheck) and h750_model (agent p1check), which agree on the hexagon sets of four cells.

Scope

every cell

Depends on

nothing beyond its definitions

Used by

Law C, law \(P^k\), Lemmas 1 to 3 (the cage laws) proved theorem-h-orientation/cage-laws
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
\(K_u\) (a uniform chain forces a direction to be absent) certified theorem-h-orientation/ku
Lemma 21 (a state repeated periodically) proved theorem-h-orientation/lemma-21
The parametrisation (C) (completeness of the chain modes) proved theorem-h-orientation/parametrisation-c
Statement R and its box lemma (Conjecture \(R_b\); Theorem U section, item 8) certified theorem-h-orientation/statement-r-box-lemma
Frozen zero-flux states with no translation symmetry on thin cells refuted theorem-h-orientation/thin-cells-aperiodic
Theorem M′ (from the project’s earlier record) proved theorem-h-orientation/thm-m-prime
Theorem U′ certified at the cubic sides four, five and six certified theorem-h-orientation/u-prime-certificates-2026-09-27
Theorem U′ certified at sides seven and eight and on four cells that are not cubic; the counterexample search certified theorem-h-orientation/u-prime-certificates-2026-10-01
The reach of (V) (Theorem U section, item 1) solver theorem-h-orientation/v-reach-law

Provenance

Written by: Lyndon Drake; Claude (Fable and Opus 5, several agents)
Checked by: Claude Opus 5 (agents pacheck, p1check): independent models from the definitions
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)