theorem-h-orientation / closed-kink-counts
The kink counts per class cannot close (V) (Theorem U section, item 10)
Closed route: counts per class cannot prove (V); what matters is which positions are kinked, not how many.
By solver on four cells with controls, the pattern of (V) makes the defect one constant on every chain of a three-grid window, non-negative on \([-3, 2g - 2]\), and with the negation zero on \([-3, 2g - 1]\) and not before, so given the pattern on the torus (V) is equivalent to a positive defect, with the same reach \(2g + 1\). Two parallel unit sheets of one sense through the two rods with defect zero satisfy the wave equation, the summed law \(P^k\), Lemma 6, the bounds, the pins of the pattern and the negation round the whole torus of each cell, and every enumerated solution short of the threshold extends by the wave equation round the rod axis within the bounds. The obstruction lies in which positions inside a class are kinked, not in how many.
- Identifier
theorem-h-orientation/closed-kink-counts- Kind
- negative-result a route shown not to work, recorded so it is not retried.
- Status
- closed-route a method shown not to reach the target.
Solver with controls (23 batches, four cells) and an exact witness (h767) on four cells; the main loop re-ran the witness on two cells and the defect queries on one window.
- Scope
- (12,4,4), (10,5,5), (12,6,6), (12,4,8)
- Depends on
- 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
- The reach of (V) (Theorem U section, item 1) solver theorem-h-orientation/v-reach-law
- Used by
- no claim in a published manifest
- Artefacts
- 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 - Notes
Do not re-run: enumerations of models for the kink vectors; window sweeps for the defect. Not tried: the counts per pair of classes of the two chain families, on which the rigid sheet is a graph.
- 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)