theorem-h-orientation / thm-u

Theorem U

In an odd frozen state every octahedral hole is oriented, one rod pointing in and one out; the one open step to Theorem H.

In an odd frozen state every octahedral hole is oriented: one of its rods points towards it and the other away. Equivalent forms: an odd state has no charged hole; two consecutive vertices of type \(k\) on a \(k\)-line have the same sign; a state with an unoriented cage has even curl; and the working form (V), that in a state with the four vertices of Theorem U′ the sublattice of the bridgeheads has no vertex of the rods’ type.

Identifier: theorem-h-orientation/thm-u
Kind: conjecture
Status: open a target with no verdict.
Proved on the cells with a coprime pair of extents and on the cells with two chains in every pair of planes; proved for the holes of one axis when the two transverse extents are coprime (Theorem 10) or have greatest common divisor two with the rod axis of odd extent (Theorem 11); certified on the cubic cells of side four to eight and on (3,3,9), (4,4,8), (3,3,12), (6,6,9). Open on every other cell. A whole-torus search for a counterexample on the six cheapest open cells on 2026-10-01 found none, with no near miss.
Scope: open on cells with no coprime pair of extents and more than two chains in every pair of planes, beyond those certified
Hypotheses: frozen, odd
Depends on: nothing beyond its definitions
Used by: no claim in a published manifest
Formal: lean4 Periodicity.theoremU (stated) in formal/periodicity
Provenance: Written by: Claude (Fable), main loop (statement, 2026-09-22)
Checked by: no independent checker
Read line by line by a person: no
Status reached: 2026-10-01
Notes: The one open statement between the record and Theorem H on every cell, and so between the record and the periodicity of frozen zero-flux states. Everything known about it is in the note’s section “Theorem U”, items 1 to 9, including the routes computation has closed; see the negative-result claims of this manifest.
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)