theorem-h-orientation / closed-r-then-monodromy
Statement R, a monodromy in the weights, and the heights are not a route to (V) (Theorem U section, item 9)
Closed route: Statement R, periodicities of the chain weights and height functions do not lead to (V); any proof must be integral from the start.
Statement R assumed on the whole rod line shortens the reach of (V) from \(2g + 1\) to \(2g - 1\), and the single literal that the third rod position has the rods’ type gives the same; (V) given R is false on both cut cylinders; on the grid of the bridgeheads of every cubic cell the parametrisation (C) is onto, so the real relaxation says nothing about the type set of a grid and the separability lemma has no product to act on; the weights of (C) seen along the rod axis return after \(\mathrm{lcm}(\gcd(a, c), \gcd(a, b))\) cubes, which is 4 on \((12, 8, 8)\) where (V) needs 15 grids; the \(g\) of the reach law is the modulus of the labels \(y - z\), \(y + z\) of the chains of the rod axis on the grids normal to it. Any proof of Theorem U must be integral from the first line. The branch certificates of the box lemma split into local ones, identical across \(b\), and global ones that span the closed circle and grow, with no rule found.
- Identifier
theorem-h-orientation/closed-r-then-monodromy- 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 with controls on five cells, exact computation on seven, and hand lemmas with no second reader (agent screw); exact certificates for \(b = 3\) to 5 (agent rbox); a sample of each re-run by the main loop.
- Scope
- the cells named in the reports; the lemmas on the grid for every cell with b | a and c | a
- Depends on
- Statement R and its box lemma (Conjecture \(R_b\); Theorem U section, item 8) certified theorem-h-orientation/statement-r-box-lemma
- The parametrisation (C) (completeness of the chain modes) proved theorem-h-orientation/parametrisation-c
- Used by
- no claim in a published manifest
- Artefacts
- Provenance
- Written by: Claude Fable 5.1 (agent screw); Claude Opus 5 (agent rbox)
Checked by: no independent checker
Read line by line by a person: no
Status reached: 2026-10-01 - Notes
Do not re-run: (V) given R on windows, cuts or tubes; any step in the transverse-chain weights along the rod axis; least-L1 branch certificates on a fixed vertex family.
- 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)