From b27a4d20a397358a28ef6d2509ad89e9312698d2 Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 03:48:48 -0400 Subject: [PATCH] coloring_nested_tire_graphs: rainbow-conjecture proof sketch + threshold counterexample MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 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). 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