theorem-h-orientation / thm-6-separability
Theorems 4 to 8 (line counts, two separable functions with product zero, walls, uniform chains beside a wall)
How the count of a type along a line depends on position: separable counts, walls, and uniform chains beside a wall.
In a frozen state the count of a type along a line is separable in the two transverse coordinates (Theorem 5); if every hole is oriented the count of a two-way type on a line depends on one transverse coordinate on each sublattice (Theorem 6); a \(k\)-free plane is a wall with one sign on the planes beside it (Theorem 7); the chains beside a wall are uniform (Theorem 8).
- Identifier
theorem-h-orientation/thm-6-separability- Kind
- theorem
- Status
- proved a written argument read by an independent checker with its own programs.
Written arguments read by independent checkers on 27 September 2026.
- Scope
- every cell
Hypotheses: frozen; orientation where stated - Depends on
- Law C, law \(P^k\), Lemmas 1 to 3 (the cage laws) proved theorem-h-orientation/cage-laws
- Theorem 3 (the diagonal law) proved theorem-h-orientation/thm-3-diagonal-law
- Used by
- Theorem 1 (the reduction), form of 30 September 2026, with Lemmas 22 to 25 proved theorem-h-orientation/thm-1
- Provenance
- Written by: Claude Opus 5 (agent rphand)
Checked by: Claude Opus 5 (agents check, check2)
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)