face_monochromatic_pairs: per-cycle refinement + Corollary 5.4

Empirical refinement of Lemma 5.3: h_phi is non-constant on V(K_b)
alone (not just on the union) and likewise on V(K_c) alone, in every
one of 142,812 chord-apex+Kempe colourings tested (n in [12, 20]).
This is strictly stronger than what we previously reported.

The proof of Lemma 5.3 already constructs the (F, e_1, e_2) witness
from any consecutive same-Heawood failure on either Kempe cycle
through merged -- never needing the other cycle. Pull that out into
a separate Corollary 5.4 ("Per-cycle form"), which makes the
empirical-to-conjecture path more direct.

Update Remark 5.5 to:
  - Cite Corollary 5.4 instead of the contrapositive of Lemma 5.3.
  - Replace "non-constant on V(K_b) U V(K_c)" with the per-cycle form.
  - Extend the empirical table with separate columns for K_b and K_c
    non-constancy.

Also commit experiments/check_constancy_obstruction.py, the script
that produced these refined empirical findings. It additionally
records that no single named vertex (v_n, A_i, ..., A_{i+4}) is
structurally majority or minority -- the minority rates cluster in
31-39%, ruling out a single-vertex-mismatch identity.

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

papers/face_monochromatic_pairs/experiments/check_constancy_obstruction.py

@@ -0,0 +1,224 @@
+"""For each chord-apex+Kempe colouring, dissect WHERE the constancy
+hypothesis fails:
+
+  (1) Is h_phi constant on V(K_b) alone? On V(K_c) alone? On the
+      intersection V(K_b) cap V(K_c)?
+  (2) Per colouring, distribution of (#+1, #-1) shared vertices.
+  (3) For each colouring, identify the "minority Heawood" shared
+      vertices (= those whose h differs from the more frequent value
+      on V(K_b) cup V(K_c)).  How many are there?  Are they
+      concentrated at specific structural positions (v_n, A_{i+1},
+      A_{i+3}, A_{i+4})?
+  (4) Is h(v_n) always equal to / always different from the majority?
+
+Under the constancy hypothesis, all of these distributions would be
+trivial (everything same Heawood, 0 minority vertices).  Their
+non-trivial empirical structure exposes where Path 4's "trap"
+mechanism would have to apply.
+
+Run with: sage experiments/check_constancy_obstruction.py
+"""
+import os
+import sys
+import time
+
+from sage.all import Graph
+from sage.graphs.graph_generators import graphs
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+sys.path.insert(0, HERE)
+
+from check_conj_3_8_scaled import (
+    apply_reduction,
+    proper_3_edge_colorings,
+    matches_chord_apex_kempe,
+    trace_kempe_cycle,
+    edge_idx,
+    kempe_cycle_set,
+)
+from check_heawood_on_kempe import (
+    dual_of, heawood_numbers, vertices_of_kempe,
+)
+
+
+def named_vertices(named, v_n=9999):
+    """Recover A_i, A_{i+1}, ..., A_{i+4}."""
+    def other(fs, v):
+        return next(iter(fs - {v}))
+    A_i  = other(named['side_0'], v_n)
+    A_i1 = other(named['spike'],  v_n)
+    A_i2 = other(named['side_1'], v_n)
+    A_i3, A_i4 = sorted(named['merged'])
+    return A_i, A_i1, A_i2, A_i3, A_i4
+
+
+def test_one(D):
+    D.is_planar(set_embedding=True)
+    n_col = 0
+    rec = {
+        'const_kb': 0, 'const_kc': 0, 'const_cap': 0,
+        # (#+1, #-1) on V(K_b) cup V(K_c) histogram
+        'plus_minus_dist': {},
+        # # minority Heawood vertices on V(K_b) cup V(K_c)
+        'minority_count': {},
+        # Is v_n minority? majority?
+        'v_n_pos': {'minority': 0, 'majority': 0, 'tie': 0},
+        # Position-tallies of minority for each "named" vertex
+        'minority_at': {'v_n': 0, 'A_i': 0, 'A_i1': 0, 'A_i2': 0,
+                         'A_i3': 0, 'A_i4': 0},
+        # Is h(v_n) == h(A_{i+3}) == h(A_{i+4}) always?
+        'v_n_eq_merged_endpts': 0,
+        # Is h constant on the 6 named vertices {v_n, A_0..A_4}?
+        'const_on_named6': 0,
+    }
+    for face in D.faces():
+        if len(face) != 5: continue
+        for i_red in range(5):
+            res = apply_reduction(D, face, i_red, 9999)
+            if res is None: continue
+            H = res['H']; named = res['named']
+            H.is_planar(set_embedding=True)
+            edges, colorings = proper_3_edge_colorings(H)
+            cand = [c for c in colorings
+                    if matches_chord_apex_kempe(edges, c, named)]
+            v_n = 9999
+            A_i, A_i1, A_i2, A_i3, A_i4 = named_vertices(named, v_n)
+            for col in cand:
+                n_col += 1
+                try:
+                    h = heawood_numbers(H, edges, col)
+                except RuntimeError:
+                    continue
+                merged_idx = edge_idx(edges, named['merged'])
+                a = col[merged_idx]
+                bs = [c for c in range(3) if c != a]
+                kc_b = kempe_cycle_set(edges, col, merged_idx, (a, bs[0]))
+                kc_c = kempe_cycle_set(edges, col, merged_idx, (a, bs[1]))
+                V_b = vertices_of_kempe(edges, kc_b)
+                V_c = vertices_of_kempe(edges, kc_c)
+                V_union = V_b | V_c
+                V_cap = V_b & V_c
+                h_b_vals = {h[v] for v in V_b}
+                h_c_vals = {h[v] for v in V_c}
+                h_cap_vals = {h[v] for v in V_cap}
+                if len(h_b_vals) == 1: rec['const_kb'] += 1
+                if len(h_c_vals) == 1: rec['const_kc'] += 1
+                if len(h_cap_vals) == 1: rec['const_cap'] += 1
+                plus = sum(1 for v in V_union if h[v] == 1)
+                minus = sum(1 for v in V_union if h[v] == -1)
+                rec['plus_minus_dist'][(plus, minus)] = \
+                    rec['plus_minus_dist'].get((plus, minus), 0) + 1
+                # Majority sign on union
+                if plus > minus:
+                    maj = 1
+                elif minus > plus:
+                    maj = -1
+                else:
+                    maj = 0
+                minority = sum(1 for v in V_union if h[v] != maj and maj != 0)
+                if maj == 0:
+                    minority = min(plus, minus)
+                rec['minority_count'][minority] = \
+                    rec['minority_count'].get(minority, 0) + 1
+                # v_n's position
+                hv = h.get(v_n)
+                if hv is not None and maj != 0:
+                    if hv == maj:
+                        rec['v_n_pos']['majority'] += 1
+                    else:
+                        rec['v_n_pos']['minority'] += 1
+                else:
+                    rec['v_n_pos']['tie'] += 1
+                # Named vertices: minority count
+                for name, vv in [('v_n', v_n), ('A_i', A_i),
+                                  ('A_i1', A_i1), ('A_i2', A_i2),
+                                  ('A_i3', A_i3), ('A_i4', A_i4)]:
+                    if vv in V_union and maj != 0 and h[vv] != maj:
+                        rec['minority_at'][name] += 1
+                # h(v_n) == h(A_{i+3}) == h(A_{i+4}) ?
+                if h.get(v_n) == h.get(A_i3) == h.get(A_i4):
+                    rec['v_n_eq_merged_endpts'] += 1
+                # h constant on the 6 named vertices?
+                named6 = [v_n, A_i, A_i1, A_i2, A_i3, A_i4]
+                named6_vals = {h[vv] for vv in named6 if vv in h}
+                if len(named6_vals) == 1:
+                    rec['const_on_named6'] += 1
+    return n_col, rec
+
+
+def merge_into(g, r):
+    for k in ('const_kb', 'const_kc', 'const_cap',
+              'v_n_eq_merged_endpts', 'const_on_named6'):
+        g[k] += r[k]
+    for k in ('plus_minus_dist', 'minority_count'):
+        for kk, vv in r[k].items():
+            g[k][kk] = g[k].get(kk, 0) + vv
+    for sub in ('v_n_pos', 'minority_at'):
+        for kk, vv in r[sub].items():
+            g[sub][kk] = g[sub].get(kk, 0) + vv
+
+
+def main(max_n=18, time_budget_per_n=1800):
+    print(f"Constancy obstruction analysis, n in [12, {max_n}]\n")
+    grand = {
+        'const_kb': 0, 'const_kc': 0, 'const_cap': 0,
+        'plus_minus_dist': {}, 'minority_count': {},
+        'v_n_pos': {'minority': 0, 'majority': 0, 'tie': 0},
+        'minority_at': {'v_n': 0, 'A_i': 0, 'A_i1': 0, 'A_i2': 0,
+                         'A_i3': 0, 'A_i4': 0},
+        'v_n_eq_merged_endpts': 0,
+        'const_on_named6': 0,
+    }
+    grand_col = 0
+    for n in range(12, max_n + 1):
+        start = time.time()
+        try:
+            triangulations = list(graphs.triangulations(n, minimum_degree=5))
+        except Exception as ex:
+            print(f"n={n}: cannot enumerate ({ex})")
+            continue
+        n_col_n = 0
+        for tri_idx, G in enumerate(triangulations):
+            if time.time() - start > time_budget_per_n:
+                print(f"  n={n}: timeout at tri {tri_idx}/{len(triangulations)}")
+                break
+            G.is_planar(set_embedding=True)
+            D = dual_of(G)
+            ni, ri = test_one(D)
+            n_col_n += ni
+            merge_into(grand, ri)
+        elapsed = time.time() - start
+        print(f"n={n}: {n_col_n} col., [{elapsed:.0f}s]")
+        sys.stdout.flush()
+        grand_col += n_col_n
+
+    print()
+    print("=" * 78)
+    print(f"Grand totals (n in [12, {max_n}], {grand_col} colourings)")
+    print(f"\n  h constant on V(K_b):             {grand['const_kb']}/{grand_col}")
+    print(f"  h constant on V(K_c):             {grand['const_kc']}/{grand_col}")
+    print(f"  h constant on V(K_b) cap V(K_c):  {grand['const_cap']}/{grand_col}")
+    print(f"  h constant on {{v_n, A_0..A_4}}:    "
+          f"{grand['const_on_named6']}/{grand_col}")
+    print(f"  h(v_n) == h(A_{{i+3}}) == h(A_{{i+4}}): "
+          f"{grand['v_n_eq_merged_endpts']}/{grand_col}")
+    print(f"\n  Minority count (= #vertices on V(K_b) cup V(K_c) with "
+          f"non-majority h) distribution:")
+    for k, v in sorted(grand['minority_count'].items()):
+        pct = 100 * v / max(1, grand_col)
+        print(f"    {k:>3}: {v} ({pct:.2f}%)")
+    print(f"\n  v_n status on V(K_b) cup V(K_c):")
+    print(f"    in majority: {grand['v_n_pos']['majority']} "
+          f"({100*grand['v_n_pos']['majority']/max(1,grand_col):.2f}%)")
+    print(f"    in minority: {grand['v_n_pos']['minority']} "
+          f"({100*grand['v_n_pos']['minority']/max(1,grand_col):.2f}%)")
+    print(f"    tie:         {grand['v_n_pos']['tie']} "
+          f"({100*grand['v_n_pos']['tie']/max(1,grand_col):.2f}%)")
+    print(f"\n  How often each named vertex is in the minority:")
+    for name, c in grand['minority_at'].items():
+        print(f"    {name:>5}: {c}/{grand_col} "
+              f"({100*c/max(1,grand_col):.2f}%)")
+
+
+if __name__ == '__main__':
+    main()

papers/face_monochromatic_pairs/paper.aux

@@ -40,10 +40,11 @@
 \newlabel{lem:both-kempe-constant}{{5.3}{11}}
 \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The two cases in the proof of Lemma\nonbreakingspace 5.2\hbox {}. Vertices $v_0, v_1$ are consecutive on the $\{a, b\}$-Kempe cycle $K$, joined by an edge $e$, with the lemma's hypothesis $h_\varphi (v_0) = h_\varphi (v_1) = +1$ --- so both vertices share the clockwise colour order $(a, b, c)$. \emph  {Left (Case\nonbreakingspace A):} when $\varphi (e) = a$, the colour-$b$ edge at $v_0$ lies south of $e$ (on $\partial F_R$) and the colour-$b$ edge at $v_1$ lies north of $e$ (on $\partial F_L$); the two would-be witness edges are on opposite faces, so no face of $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,i}$ contains both. \emph  {Right (Case\nonbreakingspace B):} when $\varphi (e) = b$, the colour-$a$ edges at $v_0, v_1$ are likewise on opposite sides of $e$. In either case the clause-$(3)$ arc of Conjecture\nonbreakingspace 5.1\hbox {} cannot be realised at $e$.}}{12}{}\protected@file@percent }
 \newlabel{fig:lemma-kempe-heawood}{{5}{12}}
-\newlabel{rem:heawood-empirical}{{5.4}{13}}
+\newlabel{cor:single-cycle-non-constancy}{{5.4}{13}}
-\newlabel{rem:conj-3-6-empirical}{{5.5}{13}}
+\newlabel{rem:heawood-empirical}{{5.5}{13}}
-\newlabel{conj:face-monochromatic-pair-strengthened}{{5.6}{14}}
+\newlabel{rem:conj-3-6-empirical}{{5.6}{13}}
-\newlabel{rem:conj-3-8-empirical}{{5.7}{14}}
+\newlabel{conj:face-monochromatic-pair-strengthened}{{5.7}{14}}
+\newlabel{rem:conj-3-8-empirical}{{5.8}{14}}
 \bibcite{Heawood1898}{1}
 \bibcite{AH77a}{2}
 \bibcite{AHK77}{3}
@@ -54,6 +55,6 @@
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
-\newlabel{rem:implication-4ct}{{5.8}{15}}
+\newlabel{rem:implication-4ct}{{5.9}{15}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{15}{}\protected@file@percent }
 \gdef \@abspage@last{15}

papers/face_monochromatic_pairs/paper.log

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 LaTeX Warning: `h' float specifier changed to `ht'.
 
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+Overfull \hbox (45.67143pt too wide) in paragraph at lines 834--848
+ [] 
+ []
 
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-\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.7\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T
+\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.8\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T
 he strength-ened con-jec-ture was tested on the same chord-
  []
 
 
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+Underfull \hbox (badness 1014) in paragraph at lines 941--947
-\OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 5.5[]; for each colour-ing we
+\OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 5.6[]; for each colour-ing we
  sought any
  []
 
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papers/face_monochromatic_pairs/paper.tex

@@ -795,43 +795,67 @@ 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}]
+\begin{corollary}[Per-cycle form]
+\label{cor:single-cycle-non-constancy}
+Let $G$, $\widehat{G}'_{v,i}$, $\varphi$ be as in
+Lemma~\ref{lem:both-kempe-constant}, and let $K$ be either of the two
+Kempe cycles of $\varphi$ through the merged edge. If $h_\varphi$ is not
+constant on $V(K)$, then a triple $(F, e_1, e_2)$ satisfying
+clauses~(1)--(3) of
+Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} on
+$(G, \widehat{G}'_{v,i}, \varphi)$ exists.
+\end{corollary}
+
+\begin{proof}
+This is precisely the case analysis used to prove
+Lemma~\ref{lem:both-kempe-constant}: applied to any consecutive pair of
+vertices on $K$ with differing Heawood numbers, the construction in
+that proof produces a clauses-(1)--(3) witness without ever needing to
+inspect the other Kempe cycle.
+\end{proof}
+
+\begin{remark}[Empirical near-proof of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} via Corollary~\ref{cor:single-cycle-non-constancy}]
 \label{rem:heawood-empirical}
 \sloppy
-The contrapositive of Lemma~\ref{lem:both-kempe-constant} reduces
+By Corollary~\ref{cor:single-cycle-non-constancy},
-Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} to
+Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
-the following structural claim:
+follows from the (a~priori weaker) structural claim:
 \emph{for every chord-apex+Kempe colouring $\varphi$ of every reduced
-dual $\widehat{G}'_{v,i}$, $h_\varphi$ is not constant on
+dual $\widehat{G}'_{v,i}$, $h_\varphi$ is not constant on $V(K_b)$
-$V(K_b) \cup V(K_c)$.} We have verified this claim computationally on
+(equivalently, not constant on $V(K_c)$).} We have verified this claim
-all chord-apex+Kempe colourings of reduced duals with $|V(G)| \le 20$
+computationally on all chord-apex+Kempe colourings of reduced duals
-(including the six Holton--McKay duals at $n = 21$ as a special case);
+with $|V(G)| \le 20$ (including the six Holton--McKay duals at
-see \texttt{experiments/check\_heawood\_on\_kempe.py}.
+$n = 21$ as a special case); see
+\texttt{experiments/check\_heawood\_on\_kempe.py} and
+\texttt{experiments/check\_constancy\_obstruction.py}.
 \begin{center}
 \small
 \renewcommand{\arraystretch}{1.15}
-\begin{tabular}{r|r|r|l}
+\begin{tabular}{r|r|r|r|l}
-$n$ & \#col.\ tested & \#non-constant on $V(K_b)\cup V(K_c)$ & status \\
+$n$ & \#col.\ tested
+    & \#non-const. on $V(K_b)$
+    & \#non-const. on $V(K_c)$ & status \\
 \hline
-$14$ & $216$       & $216$       & all non-constant \\
+$14$ & $216$       & $216$       & $216$       & all non-constant \\
-$16$ & $864$       & $864$       & all non-constant \\
+$16$ & $864$       & $864$       & $864$       & all non-constant \\
-$17$ & $4{,}650$   & $4{,}650$   & all non-constant \\
+$17$ & $4{,}650$   & $4{,}650$   & $4{,}650$   & all non-constant \\
-$18$ & $8{,}070$   & $8{,}070$   & all non-constant \\
+$18$ & $8{,}070$   & $8{,}070$   & $8{,}070$   & all non-constant \\
-$19$ & $21{,}138$  & $21{,}138$  & all non-constant \\
+$19$ & $21{,}138$  & $21{,}138$  & $21{,}138$  & all non-constant \\
-$20$ & $107{,}874$ & $107{,}874$ & all non-constant \\
+$20$ & $107{,}874$ & $107{,}874$ & $107{,}874$ & all non-constant \\
 \hline
-total ($n \le 20$) & $142{,}812$ & $142{,}812$ & \\
+total ($n \le 20$) & $142{,}812$ & $142{,}812$ & $142{,}812$ & \\
 \end{tabular}
 \end{center}
-\noindent Since $h_\varphi$ on $V(K_b) \cup V(K_c)$ was non-constant in
+\noindent In particular, $h_\varphi$ is non-constant on $V(K_b)$ alone
-every tested colouring, Lemma~\ref{lem:both-kempe-constant}'s
+in every tested colouring (and likewise on $V(K_c)$); by
-contrapositive supplies a Conjecture-\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
+Corollary~\ref{cor:single-cycle-non-constancy} each such colouring
-witness in each case --- giving an empirical near-proof of the
+admits a Conjecture-\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
-conjecture for $|V(G)| \le 20$ that is independent of (and consistent
+witness. This gives an empirical near-proof of the conjecture for
-with) the direct witness-search check of
+$|V(G)| \le 20$ independent of (and consistent with) the direct
-Remark~\ref{rem:conj-3-6-empirical}. A structural proof of
+witness-search check of Remark~\ref{rem:conj-3-6-empirical}. A
-non-constancy on $V(K_b) \cup V(K_c)$ would convert this into a proof
+structural proof of non-constancy on $V(K_b)$ (or on $V(K_c)$) would
-of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
+convert this into a proof of
+Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
 proper.
 \end{remark}