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face_monochromatic_pairs: empirical near-proof of Conjecture 5.1 via Lemma 5.3

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Add Remark 5.5 immediately after Lemma 5.3's proof, recording the
empirical reduction of Conjecture 5.1 via the contrapositive of
Lemma 5.3: the conjecture follows from "h_phi is not constant on
V(K_b) U V(K_c)", and we have verified that non-constancy holds on
every one of 142,812 chord-apex+Kempe colourings up to n <= 20
(including the six Holton-McKay duals as a special case).

This is an independent empirical near-proof of Conjecture 5.1,
complementary to the direct (1)-(3) witness check in
Remark 5.6 / rem:conj-3-6-empirical. A structural proof of the
non-constancy claim would upgrade this to a proof of the
conjecture.

Also include two diagnostic scripts that informed the remark:
  - check_shared_parity.py: parity-bucket symmetry n_{0,0} = n_{1,1},
    n_{0,1} = n_{1,0} at vertices in V(K_b) cap V(K_c). 100%.
  - check_cw_parity_prediction.py: structural identity
    s_b XOR s_c = i_b XOR i_c XOR 1 holds at every shared vertex
    (263,004 / 263,004), and the simple constancy prediction matches
    exactly 50% of shared vertices per colouring with 0 perfectly
    matching colourings.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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@@ -795,6 +795,46 @@ cycles, so its two endpoints --- which lie on $V(K_b) \cap V(K_c)$ ---
force the two constants to coincide.
\end{proof}
\begin{remark}[Empirical near-proof of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} via Lemma~\ref{lem:both-kempe-constant}]
\label{rem:heawood-empirical}
\sloppy
The contrapositive of Lemma~\ref{lem:both-kempe-constant} reduces
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} to
the following structural claim:
\emph{for every chord-apex+Kempe colouring $\varphi$ of every reduced
dual $\widehat{G}'_{v,i}$, $h_\varphi$ is not constant on
$V(K_b) \cup V(K_c)$.} We have verified this claim computationally on
all chord-apex+Kempe colourings of reduced duals with $|V(G)| \le 20$
(including the six Holton--McKay duals at $n = 21$ as a special case);
see \texttt{experiments/check\_heawood\_on\_kempe.py}.
\begin{center}
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{r|r|r|l}
$n$ & \#col.\ tested & \#non-constant on $V(K_b)\cup V(K_c)$ & status \\
\hline
$14$ & $216$ & $216$ & all non-constant \\
$16$ & $864$ & $864$ & all non-constant \\
$17$ & $4{,}650$ & $4{,}650$ & all non-constant \\
$18$ & $8{,}070$ & $8{,}070$ & all non-constant \\
$19$ & $21{,}138$ & $21{,}138$ & all non-constant \\
$20$ & $107{,}874$ & $107{,}874$ & all non-constant \\
\hline
total ($n \le 20$) & $142{,}812$ & $142{,}812$ & \\
\end{tabular}
\end{center}
\noindent Since $h_\varphi$ on $V(K_b) \cup V(K_c)$ was non-constant in
every tested colouring, Lemma~\ref{lem:both-kempe-constant}'s
contrapositive supplies a Conjecture-\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
witness in each case --- giving an empirical near-proof of the
conjecture for $|V(G)| \le 20$ that is independent of (and consistent
with) the direct witness-search check of
Remark~\ref{rem:conj-3-6-empirical}. A structural proof of
non-constancy on $V(K_b) \cup V(K_c)$ would convert this into a proof
of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
proper.
\end{remark}
\begin{remark}
\label{rem:conj-3-6-empirical}
\sloppy