Investigation of the 'outer triangle absorption' hypothesis from
notes/outer_triangle_absorption.tex:
H2 (T_n alone absorbs anything to 6 perms): REFUTED. T_n=(3,k) alone
has σ_U-projection equal to all 27 elements of {1,2,3}^3.
H1 (chain does real work): TRUE, and structurally explained:
K_3-walk parity invariant (Lemma): in any proper edge 3-coloring
of C_n viewed as a closed walk in K_3, the 3 edge-traversal counts
all have the same parity (follows from each vertex's walk-degree
being even).
σ-color count parity (Corollary): σ at the full n cycle positions
has all-same-parity color counts.
Chain preserves parity (Theorem): forward propagation through SR
tire T=(m,k) maps state with parity matching k to state with parity
matching m, via σ_U + σ_D = σ_total with parities adding mod 2.
Outer-triangle absorption (Main Theorem): at L_n with |L_n|=3,
state has all-odd color counts summing to 3, forcing each count =
1, i.e., σ is a permutation of {1,2,3}.
Empirically verified: 0 parity violations across all chain states
in 3 representative chains (sizes 30-14643).
What's left:
- Non-emptiness: state at L_n EQUALS (not just ⊆) the 6 permutations.
Empirically yes. Likely via S_3-invariance argument.
- SR-correctness for actual G (the modeling gap, not addressed here).
If non-emptiness and SR-correctness are closed, this is a structural
proof of 4CT under the PDS framework — fundamentally different from
Birkhoff/Heesch reducibility.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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