dual_decomposition: verify strengthened conjecture on 6 Holton-McKay duals

- New experiment experiments/check_conj_on_holton_mckay.py parses
  McKay's planar_code file of the 6 non-Hamiltonian 38-vertex cubic
  plane graphs (Holton-McKay) and tests both clauses (1)-(3) and (1)-(4)
  of the face-monochromatic-pair conjecture on each. Result: 17,280
  candidate colourings, all 17,280 satisfy both conjectures.
- Add a "Targeted check on the Holton-McKay duals" paragraph to
  Remark 4.4 with a per-graph table.
- Fix a latent bug in check_conj_3_8_scaled.py: b was hardcoded to
  cyc_b, leaving b == a when phi(e_1) == cyc_b (and consequently c
  ambiguous). Now correctly computes b = whichever of cyc_a/cyc_b is
  not a, raising if neither matches. The bug never crashed n <= 20
  because any() short-circuited on correctly-built witnesses; the
  Holton-McKay reductions hit it on the first witness, surfacing it.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/dual_decomposition_minimal_counterexamples/paper.tex

@@ -745,6 +745,31 @@ Conjecture-\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}-witnesses
 individually satisfy clause~(4) on each colouring, but in every case
 \emph{some} witness does. Clause~(4) is therefore an existential statement
 at the witness level, not a property of every witness.
+
+\smallskip\noindent\emph{Targeted check on the Holton--McKay duals.}
+The six $21$-vertex triangulations whose duals are the
+non-Hamiltonian $38$-vertex cubic plane graphs of Holton and McKay
+(the smallest examples falsifying Tait's conjecture) are a particularly
+interesting subfamily at $n = 21$. Running the strengthened test directly
+on these six (see
+\texttt{experiments/check\_conj\_on\_holton\_mckay.py}) gives:
+
+\begin{center}
+\small
+\renewcommand{\arraystretch}{1.15}
+\begin{tabular}{r|r|r|r|l}
+HM\# & \#pentagonal faces & \#col.\ tested & \#sat.\ (1)--(4) & status \\
+\hline
+$0$ & $10$ & $2{,}880$ & $2{,}880$ & all pass \\
+$1$ & $11$ & $2{,}880$ & $2{,}880$ & all pass \\
+$2$ & $10$ & $2{,}880$ & $2{,}880$ & all pass \\
+$3$ & $10$ & $2{,}880$ & $2{,}880$ & all pass \\
+$4$ & $11$ & $2{,}880$ & $2{,}880$ & all pass \\
+$5$ & $11$ & $2{,}880$ & $2{,}880$ & all pass \\
+\hline
+total & --- & $17{,}280$ & $17{,}280$ & \\
+\end{tabular}
+\end{center}
 \end{remark}
 
 \begin{remark}[The implication to the Four Colour Theorem]