From 83df77119986658231ebe156e242073540ce2546 Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 11:12:16 -0400 Subject: [PATCH] coloring_nested_tire_graphs: add note explaining 2-SAT solvability MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Standalone explanation of what "2-SAT solvability" means in the context of the rainbow proof (rainbow_proof.tex, Conjecture 1.5): - Defines 2-SAT in general (boolean variables, 2-variable clauses, P-time decidable). - Maps it onto our rainbow proof: variables = orientation bits o_j at D-positions; clauses = inter-D-position gap constraints; cyclic chain wraps around T'_ann. - "Solvable" ⇔ proper edge 3-coloring with given σ exists, i.e. σ ∈ π_D. - Cyclic 2-SAT can in principle fail; toy example (3-cycle of not-equal clauses = odd-cycle 2-coloring obstruction). - Empirically our system never fails for σ ∈ P_m (6-18 satisfying orientations per σ at m=6), but a structural proof is open. - Why it matters: proving Conjecture 1.5 upgrades the rainbow proof's provisional corollary into a theorem and reduces chain pigeonhole to the perms-per-half overlap. Note: two_sat_solvability.tex (3 pages). Co-Authored-By: Claude Opus 4.7 --- .../notes/two_sat_solvability.aux | 2 + .../notes/two_sat_solvability.log | 300 ++++++++++++++++++ .../notes/two_sat_solvability.pdf | Bin 0 -> 159662 bytes .../notes/two_sat_solvability.tex | 151 +++++++++ 4 files changed, 453 insertions(+) create mode 100644 papers/coloring_nested_tire_graphs/notes/two_sat_solvability.aux create mode 100644 papers/coloring_nested_tire_graphs/notes/two_sat_solvability.log create mode 100644 papers/coloring_nested_tire_graphs/notes/two_sat_solvability.pdf create mode 100644 papers/coloring_nested_tire_graphs/notes/two_sat_solvability.tex diff --git a/papers/coloring_nested_tire_graphs/notes/two_sat_solvability.aux b/papers/coloring_nested_tire_graphs/notes/two_sat_solvability.aux new file mode 100644 index 0000000..d490c35 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/two_sat_solvability.aux @@ -0,0 +1,2 @@ +\relax +\gdef \@abspage@last{2} diff --git 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