face_monochromatic_pairs: demote Theorem 5.5 to a (disproved) Conjecture

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- Proof attempt re-tagged as "Partial proof attempt (now superseded)";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>

papers/face_monochromatic_pairs/COMMENTARY.md

@@ -107,6 +107,36 @@ or revealed why they don't yield a contradiction on their own:
   `experiments/check_min_flip_structure.py`,
   `experiments/check_minority_location.py`.
 
+## Failed proof route via edge-sharing Kempe constancy
+
+A natural-looking strategy to prove Conjecture 5.1 was:
+
+> **Conjecture (now disproved):** If $K_0$ is an $\{a,b\}$-Kempe cycle
+> of $\varphi$ and $K_1$ is an $\{a,c\}$-Kempe cycle of $\varphi$ that
+> shares an edge with $K_0$, then $h_\varphi$ cannot be constant on
+> both $V(K_0)$ and $V(K_1)$ simultaneously.
+
+Combined with Lemma 5.3 (no clause-3 witness $\Rightarrow$ constancy
+on both $V(K_b)$ and $V(K_c)$), this would have closed Conjecture 5.1.
+It is **false** by a concrete counterexample (Figure in `paper.tex` at
+`\ref{fig:no-two-constant-kempe-counterexample}`). A partial proof
+attempt is preserved in the paper alongside the disproof:
+
+- Step 1 (local CW structure) — unconditional.
+- Step 2 (forced odd-crossing $\Rightarrow |E(K_0) \cap E(K_1)|$ even
+  and $\geq 2$) — unconditional.
+- Step 3 (Heawood face-sum mod 3) — unconditional but does not yield
+  a contradiction on its own.
+- Step 4 (lune-face Case A) — closes the sub-case where two shared
+  a-edges are consecutive on \emph{both} cycles (automatic when
+  $|E(K_0) \cap E(K_1)| = 2$).
+- Step 5 (general case) — open / now known false via the
+  counterexample.
+
+The counterexample shows the general case of the conjecture is
+unsalvageable; the search for a structural proof of Conjecture 5.1
+will need a different angle.
+
 ## Snapshot date
 
 This commentary is current as of commit `e688037`.