Adds a side-by-side comparison of the two proof attempts now in
the repo:
Approach 1 (cyclic 2-SAT, in rainbow_proof.tex):
Proves π_D = P_m (perms-per-half) for one antipodal-chord SP
tire when m_1 ≥ m - 1. Open piece: 2-SAT solvability
(Conjecture 1.5).
Approach 2 (König lift, in worst_case_proof_sketch.tex):
Proves |S_1 ∩ S_2| ≥ 6 for two adjacent SP tires sharing γ
when both chords are on γ. Open piece: T_2 induces a
γ-face partition (Conj t2-induces-partition).
Assessment: Approach 2 is more promising because (a) the hard step
is already proven (König's theorem), (b) it proves exactly what we
need (chain-pigeonhole non-emptiness, not the full π_D
characterisation), and (c) it directly explains the empirical
worst-case |S_1 ∩ S_2| = 6 = single S_3-orbit phenomenon.
Approach 1 still has value if we need finer control over π_D's
shape, but for just establishing non-empty overlap Approach 2
suffices.
Both approaches witness the SAME canonical 6-element worst-case
intersection (the rainbow S_3-orbit at γ=6 = the König-lifted
Latin S_3-orbit).
Recommended next move: attack Conj t2-induces-partition. Write
down the candidate induced γ-partition explicitly, verify it
computationally, then prove inclusion via transfer-matrix / fibre
lifting.
Note: two_approaches_comparison.tex (3 pages).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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