Adds a "Looser chain pigeonhole hypothesis" section to
cut_depth_label.tex, between Cut tires and the
"Connection to chain pigeonhole" section.
The conjecture says: for every non-trivial cut tire T, the joint
projection π(T) is non-empty, S_3-closed, and contains at least
one full S_3-orbit (so |π(T)| ≥ 6). Chain composition through
T_1 → T_2 → ... preserves S_3-symmetry, so R_i contains a full
S_3-orbit, and R_0 ∩ R_1 contains a common orbit, contradicting G'
being a counterexample.
Status:
- Per-tire half: provable via Prop 1.13 of paper.tex (2^n + 2(-1)^n
colorings) + S_3-equivariance.
- Chain half: open. Requires showing per-tire S_3-orbit structure
composes coherently through the depth chain.
Weaker than the rainbow / König-lift conjectures (no requirement on
specific face boundary structure), with stronger empirical support
across all cut tires tested.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
e173b1d2d4