From 56ebf49b4850972922249f205fa7a4430b8d7255 Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 04:18:00 -0400 Subject: [PATCH] coloring_nested_tire_graphs: rainbow theorem proof (sharp threshold + 2-SAT reduction) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Replaces the earlier proof sketch with a clearer attempt that: 1. Corrects the sharp threshold: m_1 >= m - 1 (not m_1 >= m). Empirically verified for m ∈ {4, 6} across m_1 ≥ m - 1. 2. Proves the ⊆ direction cleanly: each O-face dual vertex has degree m/2 ≤ 3 in T'_{f'}, so its incident spokes must be pairwise distinct, putting σ in the "perms-per-half" set P_m. 3. Reduces the ⊇ direction to a cyclic 2-SAT solvability claim (Conjecture 2sat): for each σ ∈ P_m, find an "orientation" o ∈ {0,1}^m at the m D-positions such that each length-1 gap has R_j = L_{j+1} and each length-2 gap has R_j ≠ L_{j+1}. 4. Acknowledges the gap: a naive "all-zero orientation" fails (e.g. rainbow at m_1 = 6 has the all-zero attempt fail at gap (4,5)). A satisfying assignment exists in every tested case (6-18 per σ at m=6, m_1=6) but a clean general proof awaits. Two routes outlined: S_3-equivariant case analysis, or global implication-graph analysis. 5. Confirms sharpness with explicit forcing-propagation counterexample at m=6, m_1=4 (rainbow not in π_D). 6. States provisional corollary: π_D = P_m at m_1 ≥ m - 1 (conditional on Conjecture 2sat); chain pigeonhole reduces to π_U meeting P_m. Honest about what's proven (the ⊆ half, the 2-SAT reduction, the sharpness counterexample) and what's left (the 2-SAT solvability proof). Note: rainbow_proof.tex (4 pages). Co-Authored-By: Claude Opus 4.7 --- .../notes/rainbow_proof.aux | 10 + .../notes/rainbow_proof.log | 308 ++++++++++++++++++ .../notes/rainbow_proof.pdf | Bin 0 -> 221653 bytes .../notes/rainbow_proof.tex | 279 ++++++++++++++++ 4 files changed, 597 insertions(+) create mode 100644 papers/coloring_nested_tire_graphs/notes/rainbow_proof.aux create mode 100644 papers/coloring_nested_tire_graphs/notes/rainbow_proof.log create mode 100644 papers/coloring_nested_tire_graphs/notes/rainbow_proof.pdf create mode 100644 papers/coloring_nested_tire_graphs/notes/rainbow_proof.tex diff --git a/papers/coloring_nested_tire_graphs/notes/rainbow_proof.aux b/papers/coloring_nested_tire_graphs/notes/rainbow_proof.aux new file mode 100644 index 0000000..91dd850 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/rainbow_proof.aux @@ -0,0 +1,10 @@ +\relax +\newlabel{thm:rainbow}{{1}{1}} +\newlabel{lem:o-face}{{2}{1}} +\newlabel{lem:gap}{{3}{2}} +\newlabel{prop:reduction}{{4}{2}} 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