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 <noreply@anthropic.com>
didericis
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