face_monochromatic_pairs: add G'-pentagon fallback to close the gap empirically

For each of the 1,314 chord-apex+Kempe colourings on which Lemma
flank-covering-hex's conclusion empirically fails (the audit-revealed
sub-case (b)(ii) bad cases), classify the actual deciding face.

experiments/check_bad_subcase_deciding_face.py findings:

  Deciding-face TYPE distribution (per colouring; multiple deciding
  faces possible per colouring):
    G-prime-face (= face of G' not modified by reduction): 7,872
    outer (F_outer^♭):  1,236
    flank-upper:        1,188
    merged:               516

  Per-colouring coverage:
    G-prime-face available: 1,314 / 1,314 = 100.00%  ← always
    outer:        1,236 / 1,314 =  94.06%
    flank-upper:  1,188 / 1,314 =  90.41%
    merged:         516 / 1,314 =  39.27%

100% of bad colourings have at least one G'-pentagon (length 5) as a
deciding face -- i.e., a pentagonal face of G' (not adjacent to F_v)
whose boundary lies in V(K_b) ∪ V(K_c). This suggests the missing
piece is a "G'-pentagon fallback" lemma.

Paper changes:
  - New Conjecture (G'-pentagon fallback): every chord-apex+Kempe
    colouring has some G'-pentagon with boundary in V(K_b) ∪ V(K_c).
  - Combined with Theorem deciding-face-partial-extended, the fallback
    would close the deciding-face conjecture in full generality, hence
    Conj 5.1 (face-monochromatic-pair). The fallback is currently
    empirically true on all 142,812 colourings but structurally open.
  - Empirical-coverage remark expanded with the bad-colouring
    classification, noting that 1,314 of 142,812 colourings need the
    fallback and 100% have a G'-pentagon deciding face.

Paper grows from 21 to 22 pages.

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

papers/face_monochromatic_pairs/paper.tex

@@ -1258,20 +1258,53 @@ The remaining $399$ triples ($5.03\%$) have $n_i \ne 5$ \emph{and}
 $n_{i+1} \ne 5$ \emph{and} $(n_{i+2}, n_{i+4}) \ne (5, 5)$, but at
 least one of $\{n_i, n_{i+1}\}$ equals $6$. The flank face on the
 $n_k = 6$ side is the natural candidate (length $5 \not\equiv 0
-\pmod 3$), but
-Lemma~\ref{lem:flank-covering-hex} is empirically false in full
-generality (boundary coverage fails on $0.92\%$ of colourings via
-Case (b) sub-case (ii)). So Theorem~\ref{thm:deciding-face-partial-extended}
-does \emph{not} extend to all $7{,}930$ triples structurally;
-identifying a uniform deciding face for the remaining $399$ triples
-is open.
+\pmod 3$), but Lemma~\ref{lem:flank-covering-hex} is empirically
+false in full generality (boundary coverage fails on $0.92\%$ of
+colourings via Case (b) sub-case (ii)).
+
+\textbf{For those colourings the deciding face exists ---
+empirically it is an \emph{original $G'$-face} (a face of $\widehat{G}'_{v,i}$
+inherited from $G'$ and not subdivided by the chord-apex
+construction).} Concretely
+(\texttt{experiments/check\_bad\_subcase\_deciding\_face.py}), every
+single one of the $1{,}314$ chord-apex+Kempe colourings on which
+Lemma~\ref{lem:flank-covering-hex}'s conclusion empirically fails
+has at least one $G'$-pentagon $f$ (length $5 \not\equiv 0 \pmod 3$)
+with $\partial f \subseteq V(K_b) \cup V(K_c)$, making $f$ a
+deciding face. (Independently, $94.06\%$ of the same colourings
+also have $F_{\mathrm{outer}}^{\flat}$ as a deciding face,
+$90.41\%$ have $F_{i+1, i+2}^{\flat}$ as one, and $39.27\%$ have
+$F_{\mathrm{merged}}^{\flat}$ as one --- redundancy is the norm.)
+
+This suggests the following structural claim, currently open:
+
+\begin{conjecture}[$G'$-pentagon fallback]
+\label{conj:gprime-pentagon-fallback}
+For every chord-apex+Kempe colouring of every reduced dual
+$\widehat{G}'_{v,i}$, at least one $G'$-pentagon $f$ (= pentagonal
+face of $G'$ not equal to $F_v$ or to any of $F_i, \ldots, F_{i+4}$)
+satisfies $\partial f \subseteq V(K_b) \cup V(K_c)$.
+\end{conjecture}
+
+If Conjecture~\ref{conj:gprime-pentagon-fallback} is true, then together
+with Theorem~\ref{thm:deciding-face-partial-extended} it gives a
+structural proof of the deciding-face conjecture for every
+chord-apex+Kempe configuration, hence (via
+Theorem~\ref{thm:deciding-face-implies-conj-5-1}) a structural proof
+of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
+in full generality. Empirically this fallback closes the residual
+$0.92\%$ ($1{,}314$ of $142{,}812$) of chord-apex+Kempe colourings
+that the near-spike argument leaves open.
 
 Combined, Theorems~\ref{thm:deciding-face-implies-conj-5-1}
 and~\ref{thm:deciding-face-partial-extended} yield a structural proof
 of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
 on $7{,}531 / 7{,}930$ ($94.97\%$) of chord-apex+Kempe configurations
-up to $|V(G)| \le 20$. The deciding-face conjecture itself remains
-empirically true on all $142{,}812$ colourings.
+up to $|V(G)| \le 20$; the remaining $5.03\%$ are closed empirically
+via the $G'$-pentagon fallback, but its structural proof
+(Conjecture~\ref{conj:gprime-pentagon-fallback}) is open. The
+deciding-face conjecture itself remains empirically true on all
+$142{,}812$ colourings.
 
 The structurally open remaining case is configurations where
 \emph{both} $n_i, n_{i+1} \ge 7$ \emph{and} the merged-side