theorem-h-orientation / thm-1-first-form
Theorem 1, first form (27 September 2026), through K_u
An earlier, superseded form of the reduction that also needed \(K_u\).
On every cell on which K_u holds for uniform chains of all three axes, Theorem U implies Theorem H.
- Identifier
theorem-h-orientation/thm-1-first-form- Kind
- theorem
- Status
- proved a written argument read by an independent checker with its own programs.
The earlier proof, kept under its own heading in the note’s section ‘Proof of Theorem 1’. Read by independent checkers (agents check, check2) on 27 September. Replaced as the route to Theorem H by the form of 30 September, which needs no K_u.
- Scope
- every cell on which K_u holds for all three axes
Hypotheses: frozen, odd, every hole oriented - Depends on
- \(K_u\) (a uniform chain forces a direction to be absent) certified theorem-h-orientation/ku
- Theorem M′ (from the project’s earlier record) proved theorem-h-orientation/thm-m-prime
- 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
- no claim in a published manifest
- Superseded by
- Theorem 1 (the reduction), form of 30 September 2026, with Lemmas 22 to 25 proved theorem-h-orientation/thm-1
- Artefacts
- 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)