theorem-h-orientation / closed-linear-relaxation
No weighted sum of laws proves (V) at \(g \geq 2\) (Theorem U section, items 3 to 5)
Closed route: no linear combination of the local laws proves (V) once \(g \geq 2\); integrality is essential.
With the types relaxed to non-negative reals, the largest amount of the rods’ type on the plane of the bridgeheads is positive on every cell with \(g \geq 2\) (26/3, 178/9, 299/9 at cubic sides three, four, five), so the cage laws, which span the linear consequences of (T2), give no proof there; adding the signs, the bond rule and Lemma 3 lowers the optimum by less than one; on cells with coprime transverse extents the optimum is zero. Integrality of one family of indicators suffices, and case splits leave infeasible programmes, but no structure uniform in the side was found.
- Identifier
theorem-h-orientation/closed-linear-relaxation- 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.
Computed in floating point by two implementations and verified in exact arithmetic by an independent checker on three cells.
- Scope
- cubic sides 3 to 5 and the cells named in the note
- Depends on
- Law C, law \(P^k\), Lemmas 1 to 3 (the cage laws) proved theorem-h-orientation/cage-laws
- Used by
- no claim in a published manifest
- Provenance
- Written by: Claude Opus 5 (agents of 22 and 27 September)
Checked by: Claude Opus 5 (agent check3), exact arithmetic
Read line by line by a person: no
Status reached: 2026-09-27 - Notes
Do not re-run: windows, slabs, tubes, linear programmes, thin integral families, case splits on vertices, lifts of degree two, for (V).
- 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)