theorem-h-orientation / thin-cells-aperiodic
Frozen zero-flux states with no translation symmetry on thin cells
Counterexamples: on six thin cells (an extent of 1) some frozen zero-flux states have no translation symmetry.
On the cells \((1,1,2)\), \((1,2,2)\), \((1,2,3)\), \((1,3,3)\), \((1,3,4)\) and \((1,4,5)\) there are frozen zero-flux states of even curl, with all six directions, fixed by no nonzero translation. On \((1,1,1)\) there is none.
- Identifier
theorem-h-orientation/thin-cells-aperiodic- Kind
- fact
- Status
- refuted shown false, with a witness.
Solver witnesses, each re-checked by an independent program (ice, frozen, flux zero, trivial stabiliser); the \((1,2,2)\) witness run again by the main loop. This refutes the periodicity conclusion on thin cells and shows that Theorem P’s hypothesis that every extent is at least two cannot be dropped.
- Scope
- the six thin cells named; (1,1,1) excluded
- Depends on
- The model (first section; Holes, cages and rods) proved theorem-h-orientation/model
- Used by
- no claim in a published manifest
- Artefacts
- report
reports/2026-10-01-opus-pacheck.md - script
experiments/08_frozen_structure/scratch_h/h737_thin.py(in the private repository; not published)Regenerate:cd experiments/08_frozen_structure/scratch_h && h737_thin.py 1,2,2 1,1,1 2,2,2 --cap 150
- report
- Formal
- lean4
Periodicity.not_periodic_122(proved) informal/periodicityaxioms:propext,Quot.sound - Provenance
- Written by: Claude Opus 5 (agent pacheck)
Checked by: no independent checker
Read line by line by a person: no
Status reached: 2026-10-01 - Notes
The word refuted applies to the sentence ‘every frozen zero-flux state of such a cell is invariant under a nonzero translation’ as the note’s first draft stated it for cells with a coprime pair; the corrected statement excludes extents of one.
- 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)