After several rounds of structural attempts and audits, the picture
has stabilized. Updated COMMENTARY.md with a "Final status" section:
- What we have proved: the Heawood-face-sum reduction (tight), the
tight structural cases (a', b', c) covering 94.97% of (G, v, i)
configurations, and the refined pigeonhole + S-cycle arguments
closing most of the residual ~5%.
- What's open structurally: full G'-pentagon fallback, and the
Kempe-cycle structural regularity "|S|=8 + hit=8 ⇒ p_G=11"
(30/30 empirical).
- Empirical closure: 100% across 142,812 chord-apex+Kempe colourings
up to |V(G)| ≤ 20.
- Why this isn't Appel-Haken in disguise: ~4 main steps + ~8
case-style sub-lemmas, vs RSST's 633. The chord-apex+Kempe
restriction does most of the work upfront.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Updated COMMENTARY.md with a "Lessons from the structural-proof
attempts" section summarizing:
- What worked: the Heawood-face-sum reduction (Theorem
deciding-face-implies-conj-5-1) and tight covering for n_k = 5
configurations.
- What didn't: n_i = 6 lemma (retracted as empirically false),
winding-number invariant (Σ = 0 under alternation, not
contradictory), case-analysis past |S| ≤ 1 (becomes discharging).
- Diagnostic: every global aggregation we've tried gives a quantity
consistent with constancy, not contradicting it. Lemma 5.2's
alternation is the right local consequence but doesn't aggregate
to a contradiction by itself.
- Open: G'-pentagon fallback structural proof; possible stronger
consequences of minimality of G beyond chord-apex.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
The user produced a concrete counterexample (whiteboard photo) showing
that h_φ can be constant on both an {a,b}-Kempe cycle K_0 and an
{a,c}-Kempe cycle K_1 sharing a colour-a edge.
Changes:
- theorem → conjecture environment, header marked **FALSE**
- New Remark records the disproof and identifies which step of the
proof attempt breaks: in the counterexample, no pair of shared
a-edges is consecutive on both cycles, so the lune-face premise
(Step 4 / Case A) doesn't apply
- Proof attempt re-tagged as "Partial proof attempt (now superseded)";
Steps 1-2 remain unconditional, Step 4 closes the sub-case where
some shared-a-edge pair is consecutive on both K_0 and K_1 (e.g.
automatically when |E(K_0) ∩ E(K_1)| = 2)
- Figure placeholder added referencing
figures/no-two-constant-kempe-counterexample.{png,pdf}
- COMMENTARY.md updated with a "Failed proof route" section so future
readers don't retread this path
Impact on Conjecture 5.1: the "Theorem 5.5 + Lemma 5.3 → 5.1" route
is closed; a structural proof of Conjecture 5.1 needs a different
angle. Lemma 5.3, Corollary 5.4, and the 142,812/142,812 empirical
near-proof all stand.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Standalone commentary document for readers of the paper:
- Headline table mapping each empirical / structural claim to its
proof status and the verification numbers we have.
- Statement of "what's actually open": the structural proof of
non-constancy of h_phi on V(K_b) (alone), which reduces to
Conjecture 5.1 via Corollary 5.4.
- Three reasons the proof appears to be hard:
(1) the obstruction has no slack (min flip count 2 -> 1 minority
vertex);
(2) the minority is not anchored to a structural vertex (~half
live on "other" non-named vertices);
(3) no single named-vertex-pair is always a mismatch (max 75%).
- List of candidate mechanisms ruled out by diagnostics:
- global sum identity, per-cycle sum identity,
- cycle-side balance |L| == |R|,
- specific-pair-always-mismatches.
- Index of diagnostic scripts in experiments/.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis