Attempts to prove the antipodal-chord rainbow conjecture; result is
nuanced:
1. Upper bound (proven cleanly): pi_D ⊆ {σ : (σ_0..σ_{m/2-1}) and
(σ_{m/2}..σ_{m-1}) are both perms of {1,2,3}}. This follows
from the proper-coloring constraint at the two O-face dual
vertices, each of degree m/2 in T'_{f'}.
2. Lower bound at m_1 ≥ m - 1 (constructive): every σ in the
above set extends to a proper edge 3-coloring of T'_{f'}.
Explicit construction at m=6, m_1=6. In particular rainbow
⊂ pi_D.
3. Counterexample at m_1 ≤ m - 2 (refutes original conjecture):
at m=6, m_1=4, the rainbow σ = (1,2,3,2,3,1) is NOT in pi_D.
Explicit forcing-propagation contradiction: two length-1
inter-D-position gaps on T'_ann force conflicting cycle-edge
colors at a U-position. Empirically |pi_D| = 18 (half the
full set) at m=6, m_1 ∈ {3, 4}.
REVISED conjecture: pi_D equals the full "perm-per-face" set
(containing the rainbow orbit) iff m_1 ≥ m - 1. The threshold
m_1 ≥ m - 1 is sharp. Verified for m=4 (all m_1 ≥ 3) and m=6
(m_1 ≥ 5).
Consequence: chain-pigeonhole at γ length m reduces to a smaller
overlap condition under m_1 ≥ m - 1. The case m_1 < m - 1
remains open -- pi_D still nonempty but the rainbow orbit is
missing; structural characterization of the surviving 18-element
support not addressed.
Note: rainbow_proof_sketch.tex (3 pages).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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