theorem-h-orientation / v-reach-law
The reach of (V) (Theorem U section, item 1)
How far round a hole the frozen condition must be used to force (V): the reach grows with \(g\), and the whole transverse torus is needed.
On a cell \((a, b, c)\) with the rod axis of length \(a\) and \(g = \gcd(b, c)\), the smallest windows of grids \([-p, q]\) about the hole on which (T2) forces (V) are \((3, 2g + 1)\), \((2g - 2, 2g - 2)\) and \((2g + 1, 3)\) for \(g \geq 4\), whatever \(a\); the whole transverse torus is needed, and removing the hexagons through a single transverse grid leaves the negation satisfiable.
- Identifier
theorem-h-orientation/v-reach-law- Kind
- fact
- Status
- solver a SAT or LP verdict on named cells without a checked proof object.
Solver on twenty transverse tori with \(g\) from 1 to 8, confirmed by an independent checker with its own implementation, with and without the sign field.
- Scope
- twenty cells tested; the law is conjectured for every cell
- Depends on
- 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
- The closure of the tilted sheet carries no invariant (Theorem U section, item 7) closed-route theorem-h-orientation/closed-tilted-sheet
- Artefacts
- report
reports/2026-09-27-aniso.md
- report
- Provenance
- Written by: Claude Opus 5 (agent aniso) and the main loop
Checked by: Claude Opus 5 (agent check3)
Read line by line by a person: no
Status reached: 2026-09-27 - 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)