theorem-h-orientation / kink-counts-per-class
The kink counts per chain class: the wave equation and one defect per axis (Theorem U section, item 10)
The counts of the rods’ type per chain class obey a discrete wave equation, and every chain of an axis has the same defect.
On a grid normal to the rod axis, with \(k[n][v]\) the number of vertices of the rods’ type in the class of value \(v\) (the trace of one chain of that axis), every real solution of (T2) satisfies the wave equation \(k[n+2][v] + k[n-2][v] = k[n][v+2] + k[n][v-2]\) (law C summed over the holes of a class), whose solutions are two sheets travelling one class per cube in opposite senses; and on the torus every chain of an axis has the same defect, the difference of its kink counts on its two grids, because the defect obeys a smoothing recurrence along the rod axis whose non-constant modes die round the circle.
- Identifier
theorem-h-orientation/kink-counts-per-class- Kind
- lemma
- Status
- proved-unchecked a written argument with no second reader.
Hand proofs by one agent with no second reader; the wave equation and the summed law verified to lie in the row space of the window’s equations and exact on 488 enumerated models on four cells; the single defect checked on 5,119 frozen states of five cells on all three axes; the main loop re-ran the corpus check and two solver rows.
- Scope
- every cell
Hypotheses: a real solution of one-hot and (T2); the single defect uses the closing of the rod axis - Depends on
- Law C, law \(P^k\), Lemmas 1 to 3 (the cage laws) proved theorem-h-orientation/cage-laws
- The model (first section; Holes, cages and rods) proved theorem-h-orientation/model
- Used by
- The kink counts per class cannot close (V) (Theorem U section, item 10) closed-route theorem-h-orientation/closed-kink-counts
- Artefacts
- report
reports/2026-10-01-fable-labels.md - script
experiments/08_frozen_structure/scratch_h/h764_corpus.py(in the private repository; not published)Regenerate:cd experiments/08_frozen_structure/scratch_h && h764_corpus.py
- report
- Provenance
- Written by: Claude Fable 5.1 (agent labels)
Checked by: no independent checker
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)