theorem-h-orientation / closed-tilted-sheet
The closure of the tilted sheet carries no invariant (Theorem U section, item 7)
Closed route: just short of the reach of (V), the solutions that violate it carry no usable invariant.
One grid short of the window \((3, 2g + 1)\) the negation of (V) has exactly two solutions for the set of vertices of the rods’ type, a sheet advancing one chain label per cube, for every \(g\) from 3 to 7; on every shorter window the solutions are many and dense, and no Fourier coefficient, count or winding of the types is defined on all of them or constrained on those where it is defined; opening the transverse torus does not remove \(g\).
- Identifier
theorem-h-orientation/closed-tilted-sheet- Kind
- negative-result a route shown not to work, recorded so it is not retried.
- Status
- closed-route a method shown not to reach the target.
Solver and exact computation by two agents (sheet, moment), a sample run again by the main session; no independent checker.
- Scope
- the cells with g from 3 to 7 named in the reports
- Depends on
- The reach of (V) (Theorem U section, item 1) solver theorem-h-orientation/v-reach-law
- Used by
- no claim in a published manifest
- Artefacts
- Provenance
- Written by: Claude Opus 5 (agents sheet, moment)
Checked by: no independent checker
Read line by line by a person: no
Status reached: 2026-09-29 - Notes
Do not re-run: phases, moments or Fourier coefficients of the types on the chain labels; any form of (V) on an open or cut torus.
- 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)