diff --git a/default.tikzstyles b/default.tikzstyles
new file mode 100644
index 0000000..6227919
--- /dev/null
+++ b/default.tikzstyles
@@ -0,0 +1,12 @@
+% TiKZ style file generated by TikZiT. You may edit this file manually,
+% but some things (e.g. comments) may be overwritten. To be readable in
+% TikZiT, the only non-comment lines must be of the form:
+% \tikzstyle{NAME}=[PROPERTY LIST]
+
+% Node styles
+\tikzstyle{dot}=[fill=white, draw=black, shape=circle]
+
+% Edge styles
+\tikzstyle{green edge}=[-, draw={rgb,255: red,0; green,175; blue,0}]
+\tikzstyle{red edge}=[-, draw=red]
+\tikzstyle{blue edge}=[-, draw=blue]
diff --git a/papers/face_monochromatic_pairs/constant_heawood_counterexample.tikz b/papers/face_monochromatic_pairs/constant_heawood_counterexample.tikz
new file mode 100644
index 0000000..a2f4be6
--- /dev/null
+++ b/papers/face_monochromatic_pairs/constant_heawood_counterexample.tikz
@@ -0,0 +1,106 @@
+\begin{tikzpicture}
+	\begin{pgfonlayer}{nodelayer}
+		\node [style=none] (0) at (-4, 4.5) {};
+		\node [style=none] (1) at (0, 4) {};
+		\node [style=none] (2) at (4.5, 4.5) {};
+		\node [style=none] (3) at (4, 0) {};
+		\node [style=none] (4) at (4.5, -4) {};
+		\node [style=none] (5) at (0, -4) {};
+		\node [style=none] (6) at (-4, -4) {};
+		\node [style=none] (7) at (-4, 0) {};
+		\node [style=none] (8) at (-2.5, 1.5) {};
+		\node [style=none] (9) at (-2, 2) {};
+		\node [style=none] (10) at (-1.5, 2.5) {};
+		\node [style=none] (11) at (-1, 3) {};
+		\node [style=none] (12) at (-0.5, 3.5) {};
+		\node [style=none] (13) at (-3, 1) {};
+		\node [style=none] (14) at (-3.5, 0.5) {};
+		\node [style=none] (15) at (0.5, 4.5) {};
+		\node [style=none] (16) at (0.5, -3.5) {};
+		\node [style=none] (17) at (1, -3) {};
+		\node [style=none] (18) at (1.5, -2.5) {};
+		\node [style=none] (19) at (2, -2) {};
+		\node [style=none] (20) at (2.5, -1.5) {};
+		\node [style=none] (21) at (3, -1) {};
+		\node [style=none] (22) at (3.5, -0.5) {};
+		\node [style=none] (23) at (4.5, 0.5) {};
+		\node [style=none] (24) at (-3.5, 1.25) {};
+		\node [style=none] (25) at (-2.5, 2.25) {};
+		\node [style=none] (26) at (-1.5, 3.25) {};
+		\node [style=none] (27) at (-0.5, 4.25) {};
+		\node [style=none] (28) at (-2, 0) {};
+		\node [style=none] (29) at (0, -2) {};
+		\node [style=none] (30) at (2, 0) {};
+		\node [style=none] (31) at (0, 2) {};
+		\node [style=none] (32) at (-1, 1) {};
+		\node [style=none] (33) at (1, 3) {};
+		\node [style=none] (34) at (1, -1) {};
+		\node [style=none] (35) at (3, 1) {};
+		\node [style=none] (36) at (1.25, -3.5) {};
+		\node [style=none] (37) at (2.25, -2.5) {};
+		\node [style=none] (38) at (3.25, -1.5) {};
+		\node [style=none] (39) at (4.25, -0.5) {};
+	\end{pgfonlayer}
+	\begin{pgfonlayer}{edgelayer}
+		\draw [style=blue edge] (0.center) to (15.center);
+		\draw [style=blue edge] (2.center) to (23.center);
+		\draw [style=blue edge] (5.center) to (4.center);
+		\draw [style=blue edge] (7.center) to (6.center);
+		\draw [style=red edge] (0.center) to (7.center);
+		\draw [style=red edge] (15.center) to (2.center);
+		\draw [style=red edge] (23.center) to (4.center);
+		\draw [style=red edge] (5.center) to (6.center);
+		\draw [style=red edge] (14.center) to (13.center);
+		\draw [style=red edge] (8.center) to (9.center);
+		\draw [style=red edge] (10.center) to (11.center);
+		\draw [style=red edge] (12.center) to (1.center);
+		\draw [style=red edge] (16.center) to (17.center);
+		\draw [style=red edge] (18.center) to (19.center);
+		\draw [style=red edge] (20.center) to (21.center);
+		\draw [style=red edge] (22.center) to (3.center);
+		\draw [style=green edge] (7.center) to (14.center);
+		\draw [style=green edge] (13.center) to (8.center);
+		\draw [style=green edge] (9.center) to (10.center);
+		\draw [style=green edge] (11.center) to (12.center);
+		\draw [style=green edge] (1.center) to (15.center);
+		\draw [style=green edge] (5.center) to (16.center);
+		\draw [style=green edge] (17.center) to (18.center);
+		\draw [style=green edge] (19.center) to (20.center);
+		\draw [style=green edge] (21.center) to (22.center);
+		\draw [style=green edge] (3.center) to (23.center);
+		\draw [style=green edge, in=-135, out=-45] (6.center) to (4.center);
+		\draw [style=green edge, bend left=45] (0.center) to (2.center);
+		\draw [style=blue edge] (24.center) to (14.center);
+		\draw [style=blue edge] (25.center) to (8.center);
+		\draw [style=blue edge] (26.center) to (10.center);
+		\draw [style=blue edge] (27.center) to (12.center);
+		\draw [style=red edge] (24.center) to (25.center);
+		\draw [style=red edge] (26.center) to (27.center);
+		\draw [style=green edge] (25.center) to (26.center);
+		\draw [style=green edge, bend left] (24.center) to (27.center);
+		\draw [style=blue edge] (13.center) to (28.center);
+		\draw [style=blue edge] (9.center) to (32.center);
+		\draw [style=blue edge] (11.center) to (31.center);
+		\draw [style=blue edge] (1.center) to (33.center);
+		\draw [style=blue edge] (29.center) to (17.center);
+		\draw [style=blue edge] (34.center) to (19.center);
+		\draw [style=blue edge] (30.center) to (21.center);
+		\draw [style=blue edge] (35.center) to (3.center);
+		\draw [style=blue edge] (16.center) to (36.center);
+		\draw [style=blue edge] (18.center) to (37.center);
+		\draw [style=blue edge] (20.center) to (38.center);
+		\draw [style=blue edge] (22.center) to (39.center);
+		\draw [style=red edge] (36.center) to (37.center);
+		\draw [style=red edge] (38.center) to (39.center);
+		\draw [style=green edge] (38.center) to (37.center);
+		\draw [style=green edge, bend right] (36.center) to (39.center);
+		\draw [style=red edge] (28.center) to (32.center);
+		\draw [style=red edge] (31.center) to (33.center);
+		\draw [style=red edge] (30.center) to (35.center);
+		\draw [style=red edge] (29.center) to (34.center);
+		\draw [style=green edge] (32.center) to (34.center);
+		\draw [style=green edge] (28.center) to (29.center);
+		\draw [style=green edge] (31.center) to (30.center);
+		\draw [style=green edge] (33.center) to (35.center);
+	\end{pgfonlayer}
+\end{tikzpicture}
diff --git a/papers/face_monochromatic_pairs/experiments/counterexample_conj_5_5.py b/papers/face_monochromatic_pairs/experiments/counterexample_conj_5_5.py
new file mode 100644
index 0000000..9bfaaf6
--- /dev/null
+++ b/papers/face_monochromatic_pairs/experiments/counterexample_conj_5_5.py
@@ -0,0 +1,371 @@
+"""Counterexample to Conjecture 5.5 (transcribed from
+`constant_heawood_counterexample.tikz`).
+
+This script:
+  (1) builds H from the edge list transcribed from the .tikz file;
+  (2) verifies H is simple, cubic, planar;
+  (3) verifies the colour assignment is a proper 3-edge-colouring;
+  (4) computes h_φ at every vertex via the CW rotation around each
+      vertex using the planar coordinates given in the .tikz file
+      (rather than Sage's combinatorial embedding, which may pick a
+      different rotation system);
+  (5) identifies the {a,b}-, {a,c}-, {b,c}-Kempe cycles of φ;
+  (6) reports whether h_φ is constant on each Kempe cycle;
+  (7) saves a PNG of the planar drawing with edges coloured red /
+      blue / green at the same positions as the .tikz file.
+
+Convention:
+  colour 0 = red   = "a",
+  colour 1 = blue  = "b",
+  colour 2 = green = "c".
+
+Run with: sage experiments/counterexample_conj_5_5.py
+"""
+import math
+import os
+
+from sage.all import Graph
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+OUT_PNG = os.path.join(HERE, '..', 'figures',
+                       'no-two-constant-kempe-counterexample.png')
+
+RED, BLUE, GREEN = 0, 1, 2
+COLOUR_NAME = {RED: 'red', BLUE: 'blue', GREEN: 'green'}
+
+# Vertex positions transcribed verbatim from
+# constant_heawood_counterexample.tikz.
+POS = {
+    0: (-4.0,  4.5),
+    1: ( 0.0,  4.0),
+    2: ( 4.5,  4.5),
+    3: ( 4.0,  0.0),
+    4: ( 4.5, -4.0),
+    5: ( 0.0, -4.0),
+    6: (-4.0, -4.0),
+    7: (-4.0,  0.0),
+    8: (-2.5,  1.5),
+    9: (-2.0,  2.0),
+    10: (-1.5,  2.5),
+    11: (-1.0,  3.0),
+    12: (-0.5,  3.5),
+    13: (-3.0,  1.0),
+    14: (-3.5,  0.5),
+    15: ( 0.5,  4.5),
+    16: ( 0.5, -3.5),
+    17: ( 1.0, -3.0),
+    18: ( 1.5, -2.5),
+    19: ( 2.0, -2.0),
+    20: ( 2.5, -1.5),
+    21: ( 3.0, -1.0),
+    22: ( 3.5, -0.5),
+    23: ( 4.5,  0.5),
+    24: (-3.5,  1.25),
+    25: (-2.5,  2.25),
+    26: (-1.5,  3.25),
+    27: (-0.5,  4.25),
+    28: (-2.0,  0.0),
+    29: ( 0.0, -2.0),
+    30: ( 2.0,  0.0),
+    31: ( 0.0,  2.0),
+    32: (-1.0,  1.0),
+    33: ( 1.0,  3.0),
+    34: ( 1.0, -1.0),
+    35: ( 3.0,  1.0),
+    36: ( 1.25, -3.5),
+    37: ( 2.25, -2.5),
+    38: ( 3.25, -1.5),
+    39: ( 4.25, -0.5),
+}
+
+# Edges with colours, transcribed from the .tikz edgelayer in order.
+# Two edges (6-4 and 0-2) are drawn bent in TikZ and represent the
+# "outer-face" green arcs; their straight-line geometry in POS would
+# pass through the interior, but the embedding treats them as edges of
+# the outer face (i.e. they leave each endpoint pointing away from the
+# interior). We model this by overriding the angle at vertices 0, 2,
+# 6, 4 for these two edges (see CW_ANGLE_OVERRIDE below).
+EDGES = [
+    # blue (4)
+    (0, 15, BLUE), (2, 23, BLUE), (5, 4, BLUE), (7, 6, BLUE),
+    # red (outer 4)
+    (0, 7, RED), (15, 2, RED), (23, 4, RED), (5, 6, RED),
+    # red (interior-segments group 1)
+    (14, 13, RED), (8, 9, RED), (10, 11, RED), (12, 1, RED),
+    (16, 17, RED), (18, 19, RED), (20, 21, RED), (22, 3, RED),
+    # green (outer "spokes" connecting outer frame to inner parallelograms)
+    (7, 14, GREEN), (13, 8, GREEN), (9, 10, GREEN), (11, 12, GREEN),
+    (1, 15, GREEN), (5, 16, GREEN), (17, 18, GREEN), (19, 20, GREEN),
+    (21, 22, GREEN), (3, 23, GREEN),
+    # green outer-face arcs
+    (6, 4, GREEN),   # bottom arc (bent in tikz: out=-135, in=-45)
+    (0, 2, GREEN),   # top arc (bent in tikz: bend left=45)
+    # inner upper-left "trapezoid" connecting 14, 8, 10, 12 to 24..27
+    (24, 14, BLUE), (25, 8, BLUE), (26, 10, BLUE), (27, 12, BLUE),
+    (24, 25, RED), (26, 27, RED),
+    (25, 26, GREEN),
+    (24, 27, GREEN),  # bent
+    # mid-layer edges connecting outer-frame intermediate vertices to
+    # the inner "diamond" of 28-35
+    (13, 28, BLUE), (9, 32, BLUE), (11, 31, BLUE), (1, 33, BLUE),
+    (29, 17, BLUE), (34, 19, BLUE), (30, 21, BLUE), (35, 3, BLUE),
+    # lower-right "trapezoid"
+    (16, 36, BLUE), (18, 37, BLUE), (20, 38, BLUE), (22, 39, BLUE),
+    (36, 37, RED), (38, 39, RED),
+    (38, 37, GREEN),
+    (36, 39, GREEN),  # bent
+    # inner diamond edges (red rungs)
+    (28, 32, RED), (31, 33, RED), (30, 35, RED), (29, 34, RED),
+    # inner diamond edges (green rungs)
+    (32, 34, GREEN), (28, 29, GREEN), (31, 30, GREEN), (33, 35, GREEN),
+]
+
+# For the two outer-face green arcs (and the bent edges 24-27 and
+# 36-39), override the angle used in the CW-rotation computation so
+# that the cyclic order around each endpoint matches the planar
+# embedding actually drawn in tikz rather than the straight-line
+# geometry.
+#
+# Convention: angle is measured in degrees, counterclockwise from
+# +x axis. The override gives the direction the edge LEAVES the
+# endpoint along the tikz-drawn curve, not toward the other endpoint
+# of the straight line.
+#
+# - 0->2 green arc bends "left" over the top; from 0 it leaves
+#   roughly upward (north), and from 2 it arrives from the upper-left
+#   (so 2's local angle is also roughly upward).
+# - 6->4 green arc bends "around the bottom"; from 6 it leaves
+#   roughly downward (south), and from 4 it leaves roughly downward.
+# - 24->27 green arc bends "left" above the upper-left ladder
+#   (in=-135, out=-45 conceptually).
+# - 36->39 green arc bends "right" below the lower-right ladder.
+CW_ANGLE_OVERRIDE = {
+    (0, 2):   90.0,   # at 0, edge to 2 leaves upward
+    (2, 0):   90.0,   # at 2, edge to 0 leaves upward
+    (6, 4):  -90.0,   # at 6, edge to 4 leaves downward
+    (4, 6):  -90.0,   # at 4, edge to 6 leaves downward
+    (24, 27): 135.0,  # at 24, edge to 27 leaves upper-left
+    (27, 24): 135.0,  # at 27, edge to 24 leaves upper-left
+    (36, 39): -45.0,  # at 36, edge to 39 leaves lower-right
+    (39, 36): -45.0,  # at 39, edge to 36 leaves lower-right
+}
+
+
+def angle_at(v, u):
+    """Direction (degrees) from v toward u in the planar drawing,
+    honouring CW_ANGLE_OVERRIDE for bent edges."""
+    if (v, u) in CW_ANGLE_OVERRIDE:
+        return CW_ANGLE_OVERRIDE[(v, u)]
+    (vx, vy) = POS[v]
+    (ux, uy) = POS[u]
+    return math.degrees(math.atan2(uy - vy, ux - vx))
+
+
+def build_graph(edges):
+    G = Graph(multiedges=False, loops=False)
+    for u, v, _ in edges:
+        G.add_edge(u, v)
+    return G
+
+
+def edge_colour_map(edges):
+    return {frozenset((u, v)): c for u, v, c in edges}
+
+
+def is_proper_3_edge_colouring(G, col):
+    for v in G.vertex_iterator():
+        cols = sorted(col[frozenset((v, u))] for u in G.neighbors(v))
+        if cols != [RED, BLUE, GREEN]:
+            return False, v, cols
+    return True, None, None
+
+
+def cw_neighbours(G, v):
+    """List of v's neighbours in CW order from the planar coordinates
+    (CW = decreasing polar angle)."""
+    nbrs = list(G.neighbors(v))
+    nbrs.sort(key=lambda u: -angle_at(v, u))
+    return nbrs
+
+
+def heawood_numbers(G, col):
+    h = {}
+    for v in G.vertex_iterator():
+        nbrs = cw_neighbours(G, v)
+        cols = [col[frozenset((v, u))] for u in nbrs]
+        i0 = cols.index(RED)
+        rot = cols[i0:] + cols[:i0]
+        if rot == [RED, BLUE, GREEN]:
+            h[v] = +1
+        elif rot == [RED, GREEN, BLUE]:
+            h[v] = -1
+        else:
+            raise RuntimeError(f"bad rotation at {v}: {cols}")
+    return h
+
+
+def kempe_cycle(G, col, start_edge, two_colours):
+    target = set(two_colours)
+    u0, v0 = start_edge
+    walk = [u0, v0]
+    while True:
+        cur, prev = walk[-1], walk[-2]
+        nxt = None
+        for u in G.neighbors(cur):
+            if u == prev:
+                continue
+            if col[frozenset((cur, u))] in target:
+                nxt = u
+                break
+        if nxt is None:
+            raise RuntimeError(f"Kempe walk stuck at {cur}")
+        if nxt == walk[0] and cur == walk[-2]:
+            # safety; shouldn't trigger in normal walks
+            break
+        walk.append(nxt)
+        if walk[-1] == walk[0] and len(walk) > 2:
+            return walk[:-1]
+
+
+def report(name, cycle, h, two_colours):
+    h_vals = [h[v] for v in cycle]
+    plus = sum(1 for x in h_vals if x == +1)
+    minus = sum(1 for x in h_vals if x == -1)
+    const = (plus == 0 or minus == 0)
+    a, b = two_colours
+    print(f"  {name} (colours {{ {COLOUR_NAME[a]}, {COLOUR_NAME[b]} }}):")
+    print(f"    length  = {len(cycle)}")
+    print(f"    vertices= {cycle}")
+    print(f"    h_φ     = {h_vals}")
+    print(f"    +/-     = {plus} (+1) / {minus} (-1)")
+    print(f"    const?  {'YES' if const else 'no'}")
+    return const
+
+
+# Bent edges, matching the tikz embedding. For each, we give a "rad"
+# value (positive => arc bulges to the LEFT of the directed edge u->v
+# in matplotlib's connectionstyle convention).
+BENT_EDGES = {
+    # In matplotlib's arc3, positive rad bulges to the RIGHT of the
+    # directed edge from posA to posB. We pick signs so the bulge
+    # matches the tikz drawing.
+    frozenset((0, 2)):   ('arc', (0, 2), -0.30),   # 0→2 rightward, bulge up   (left of dir)
+    frozenset((6, 4)):   ('arc', (6, 4), +0.30),   # 6→4 rightward, bulge down (right of dir)
+    frozenset((24, 27)): ('arc', (24, 27), -0.30), # 24→27 up-right, bulge up-left ("bend left")
+    frozenset((36, 39)): ('arc', (36, 39), +0.30), # 36→39 up-right, bulge down-right ("bend right")
+}
+
+
+def draw_png(G, col, out_path):
+    import matplotlib
+    matplotlib.use('Agg')
+    import matplotlib.pyplot as plt
+    from matplotlib.patches import FancyArrowPatch
+
+    os.makedirs(os.path.dirname(out_path), exist_ok=True)
+    fig, ax = plt.subplots(figsize=(10, 10), dpi=160)
+    ax.set_aspect('equal')
+    ax.axis('off')
+
+    # Draw edges first so vertices sit on top
+    for e in G.edge_iterator(labels=False):
+        u, v = e
+        c = COLOUR_NAME[col[frozenset(e)]]
+        if frozenset(e) in BENT_EDGES:
+            _, (a, b), rad = BENT_EDGES[frozenset(e)]
+            # Orient so the rad sign matches the (a -> b) direction
+            if (u, v) != (a, b):
+                rad = -rad
+            arc = FancyArrowPatch(
+                POS[u], POS[v],
+                arrowstyle='-',
+                connectionstyle=f'arc3,rad={rad}',
+                color=c, linewidth=2.5,
+                shrinkA=0, shrinkB=0,
+            )
+            ax.add_patch(arc)
+        else:
+            (x1, y1), (x2, y2) = POS[u], POS[v]
+            ax.plot([x1, x2], [y1, y2], color=c, linewidth=2.5,
+                    solid_capstyle='round')
+
+    # Draw vertices
+    for v, (x, y) in POS.items():
+        ax.plot(x, y, 'o', markersize=16,
+                markerfacecolor='lightgrey',
+                markeredgecolor='black', markeredgewidth=1.0)
+        ax.text(x, y, str(v), ha='center', va='center', fontsize=8)
+
+    # Set bounds with padding (the top arc goes well above 4.5)
+    xs = [p[0] for p in POS.values()]
+    ys = [p[1] for p in POS.values()]
+    pad = 1.5
+    ax.set_xlim(min(xs) - pad, max(xs) + pad)
+    ax.set_ylim(min(ys) - pad, max(ys) + pad)
+
+    fig.tight_layout()
+    fig.savefig(out_path, dpi=160, bbox_inches='tight')
+    plt.close(fig)
+    print(f"\nWrote: {out_path}")
+
+
+def main():
+    print("Loading edges from transcribed tikz...")
+    G = build_graph(EDGES)
+    col = edge_colour_map(EDGES)
+    print(f"  |V(H)| = {G.order()}, |E(H)| = {G.size()}")
+
+    degs = sorted(set(G.degree()))
+    print(f"  degree set = {degs}")
+    if degs != [3]:
+        print("  !! NOT CUBIC"); draw_png(G, col, OUT_PNG); return
+    print("  cubic ✓")
+
+    is_planar = G.is_planar(set_embedding=True)
+    print(f"  planar (by Sage)? {is_planar}")
+    if not is_planar:
+        print("  !! NOT PLANAR"); draw_png(G, col, OUT_PNG); return
+
+    ok, bad, cols = is_proper_3_edge_colouring(G, col)
+    if not ok:
+        print(f"  !! NOT a proper 3-edge-colouring: {bad} has {cols}")
+        draw_png(G, col, OUT_PNG); return
+    print("  proper 3-edge-colouring ✓")
+
+    h = heawood_numbers(G, col)
+    print("\nHeawood numbers per vertex (using tikz coordinates):")
+    for v in sorted(h):
+        print(f"  h_φ({v:2d}) = {h[v]:+d}")
+
+    print("\nGlobal Heawood sums:")
+    plus = sum(1 for v in h if h[v] == +1)
+    minus = sum(1 for v in h if h[v] == -1)
+    print(f"  total: {plus} (+1) / {minus} (-1), sum = {plus - minus}")
+
+    # Try all three Kempe-cycle pairs and identify the ones sharing an
+    # edge. We pick an edge of each colour and trace each cycle.
+    print("\nAll three Kempe-cycle types through some shared edge:")
+    # find an edge of each colour
+    edge_by_color = {}
+    for u, v, c in EDGES:
+        if c not in edge_by_color:
+            edge_by_color[c] = (u, v)
+
+    for a in (RED, BLUE, GREEN):
+        b, c = [x for x in (RED, BLUE, GREEN) if x != a]
+        e_a = edge_by_color[a]
+        Kab = kempe_cycle(G, col, e_a, (a, b))
+        Kac = kempe_cycle(G, col, e_a, (a, c))
+        print(f"\n-- shared edge: colour-{COLOUR_NAME[a]} edge {e_a}")
+        const_ab = report(f"K_{{a,b}}={{{COLOUR_NAME[a]},{COLOUR_NAME[b]}}}",
+                          Kab, h, (a, b))
+        const_ac = report(f"K_{{a,c}}={{{COLOUR_NAME[a]},{COLOUR_NAME[c]}}}",
+                          Kac, h, (a, c))
+        if const_ab and const_ac:
+            print(f"  ** counterexample with a={COLOUR_NAME[a]} **")
+
+    draw_png(G, col, OUT_PNG)
+
+
+if __name__ == '__main__':
+    main()
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