theorem-h-orientation / thm-3-diagonal-law
Theorem 3 (the diagonal law)
Two vertices of the same type on diagonally neighbouring lines have equal signs when the holes between them are oriented.
Two vertices of type \(k\) on diagonally neighbouring \(k\)-lines at odd separation \(s \geq 5\) with every hole of the causal diamond between them oriented have equal signs.
- Identifier
theorem-h-orientation/thm-3-diagonal-law- Kind
- theorem
- Status
- proved a written argument read by an independent checker with its own programs.
By Theorem 9’s certificate for every separation; earlier certified directly at sides 4, 5, 6 in the model with signs.
- Scope
- every cell
Hypotheses: frozen, every hole of the diamond oriented - Depends on
- Theorem 9 (the certificate of the diagonal law, closed form) proved theorem-h-orientation/thm-9-diagonal-law
- Used by
- The first step of \(K_u\) given orientation, by two applications of the diagonal law proved theorem-h-orientation/ku-first-step-hand
- Theorem 1 (the reduction), form of 30 September 2026, with Lemmas 22 to 25 proved theorem-h-orientation/thm-1
- Theorem 1, first form (27 September 2026), through K_u proved theorem-h-orientation/thm-1-first-form
- Theorems 4 to 8 (line counts, two separable functions with product zero, walls, uniform chains beside a wall) proved theorem-h-orientation/thm-6-separability
- Provenance
- Written by: Claude Opus 5 (agents lines, cone)
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)