face_monochromatic_pairs: honest audit of partial proof — flag n_i = 6 gap

Audit of the structural proof of Conjecture 5.1 (via deciding-face
conjecture) identifies one real proof gap:

Lemma (Flank covering, n_i = 6), Case (b) sub-case (ii) -- when
φ(A_i P_1) = c_1 AND φ(P_1 P_2) = c_0 -- the propagation argument
"the cycle at P_2 passes from P_2 to P_1" requires the {c, c_0}-Kempe
cycle through P_2 to be K_b, which forces P_1 onto K_b via the c_0
edge P_1 P_2. Properness at P_2 only forces P_2 ∈ V(K_c) (via
φ(A_{i+1} P_2) = c_1), not P_2 ∈ V(K_b). The further step requires
controlling the {c, c_0}-walk through the rest of the graph, which
the local argument doesn't do.

experiments/audit_tight_coverage.py quantifies the impact across
empirical data:
  - 7,930 / 7,930 (G, v, i) triples up to |V(G)| ≤ 20 are covered
    by the FULL partial proof (including the n_i = 6 lemma);
  - 7,531 / 7,930 (94.97%) are covered by the TIGHT subset
    (n_i = 5 OR n_{i+1} = 5 OR (n_{i+2}, n_{i+4}) = (5, 5)) which
    has no proof gap;
  - 399 (5.03%) genuinely require the n_i = 6 lemma.

So the gap matters: empirical coverage of the tight subset alone is
~95%, not 100%.

Paper changes:
  - Lemma (Flank covering, n_i = 6) marked as "partial" with a status
    note in the statement itself.
  - Proof of Lemma includes an "Audit note" identifying the open
    sub-case explicitly, after establishing the parts that ARE proven.
  - Empirical coverage remark softened: the 100% claim is restated
    as "modulo the open sub-case", with the 94.97% tight figure
    given separately.

Empirically the n_i = 6 lemma is robust (all 142,812 colourings have
a deciding face), so the gap is probably patchable — likely either
via a structural argument that rules out the bad sub-case in
chord-apex+Kempe colourings, or via a global K_b-walk argument
showing P_2 ∈ V(K_b) anyway. But this is open.

Paper stays at 21 pages (only added text within existing lemma + remark).

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

papers/face_monochromatic_pairs/paper.tex

@@ -1041,9 +1041,21 @@ its colour is in $\{c_0, c_1\}$, so the corresponding Kempe cycle
 placing $P$ in that cycle's vertex set.
 \end{proof}
 
-\begin{lemma}[Flank covering, $n_i = 6$]
+\begin{lemma}[Flank covering, $n_i = 6$; partial]
 \label{lem:flank-covering-hex}
 If $n_i = 6$ then $\partial F_{i, i+1}^{\flat} \subseteq V(K_b) \cup V(K_c)$.
+
+\emph{Status:} The proof below establishes the conclusion except in a
+specific sub-case (Case (b), sub-case~(ii) below) where the local
+propagation argument from $P_2$ to $P_1$ requires the
+$\{c, c_0\}$-Kempe cycle through $P_2$ to coincide with $K_b$; this
+isn't forced by the local data and would require a global argument
+through the rest of the graph. Empirically (audit of $7{,}930$
+$(G, v, i)$ triples up to $|V(G)| \le 20$,
+\texttt{experiments/audit\_tight\_coverage.py}) the gap does not
+manifest --- $399$ of the $7{,}930$ triples rely on this lemma and
+all of them have a deciding face. So the lemma's conclusion is
+empirically robust; only the structural proof is incomplete.
 \end{lemma}
 
 \begin{proof}
@@ -1084,6 +1096,29 @@ to $P_1$. If this cycle is $K_b$, $P_1 \in V(K_b)$; if it is $K_c$
 $\{c, c_1\}$), then $P_1 \in V(K_c)$.
 
 In either case $P_1 \in V(K_b) \cup V(K_c)$.
+
+\textbf{Audit note.} In Case~(b) sub-case~(ii) (where
+$\varphi(A_i P_1) = c_1$ \emph{and} $\varphi(P_1 P_2) = c_0$), the
+preceding paragraph's claim that ``the cycle at $P_2$ passes from
+$P_2$ to $P_1$'' requires the $\{c, c_0\}$-Kempe cycle through $P_2$
+to be $K_b$ (so that the $c_0$-edge $P_1 P_2$ is on the cycle).
+We can rule out one configuration of this sub-case by properness at
+$P_2$: $\varphi(A_{i+1} P_2)$ cannot be $c_0$ (since $\varphi(P_1 P_2)
+= c_0$ already accounts for $P_2$'s colour-$c_0$ edge), nor $c$ (since
+$A_{i+1}$'s only colour-$c$ edge is the spike). So
+$\varphi(A_{i+1} P_2) = c_1$, and hence $P_2 \in V(K_c)$ via the
+$K_c$-walk $A_{i+1} \to P_2$ along the $c_1$-edge.
+
+However the further conclusion $P_2 \in V(K_b)$ --- needed for the
+above argument to place $P_1 \in V(K_b)$ via the $c_0$-edge
+$P_1 P_2$ --- is not forced by the local picture; whether the
+$\{c, c_0\}$-cycle through $P_2$ coincides with $K_b$ depends on the
+$\{c, c_0\}$-walk through the rest of the graph, which we have not
+controlled here. This gap is empirically inactive on
+all chord-apex+Kempe colourings tested
+($142{,}812$ / $142{,}812$ colourings, $|V(G)| \le 20$, each with a
+deciding face), but the structural proof of this sub-case remains
+\emph{open}.
 \end{proof}
 
 \begin{theorem}[Partial proof of
@@ -1205,10 +1240,18 @@ they satisfy case (c).
 Combining
 Theorems~\ref{thm:deciding-face-implies-conj-5-1}
 and~\ref{thm:deciding-face-partial-extended}, this yields a
-\textbf{structural proof of
+structural proof of
 Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
 on every chord-apex+Kempe colouring of every reduced dual with
-$|V(G)| \le 20$.}
+$|V(G)| \le 20$, \emph{modulo} the open sub-case in
+Lemma~\ref{lem:flank-covering-hex}. A stricter accounting
+(\texttt{experiments/audit\_tight\_coverage.py}) shows that
+$7{,}531$ of the $7{,}930$ $(G, v, i)$ configurations
+($94.97\%$) are covered by the \emph{tight} subset of cases
+(\textbf{a$'$}~$n_i = 5$, \textbf{b$'$}~$n_{i+1} = 5$, \textbf{c}~$n_{i+2} = n_{i+4} = 5$),
+which has no proof gap. The remaining $399$ ($5.03\%$) rely on the
+$n_i = 6$ flank-covering lemma, and would only constitute a complete
+structural proof once that lemma's Case (b) sub-case (ii) is closed.
 
 The structurally open remaining case is configurations where
 \emph{both} $n_i, n_{i+1} \ge 7$ \emph{and} the merged-side