diff --git a/papers/face_monochromatic_pairs/experiments/check_c28_no_chord_apex_kempe_constancy.py b/papers/face_monochromatic_pairs/experiments/check_c28_no_chord_apex_kempe_constancy.py
new file mode 100644
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+++ b/papers/face_monochromatic_pairs/experiments/check_c28_no_chord_apex_kempe_constancy.py
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+"""Confirm that the C_{28} counterexample to Conjecture 5.5 (face-length-≥-5
+form) does NOT yield a counterexample to Conjecture 5.1.
+
+Concretely:
+  - The C_{28} fullerene is the dual of the 3rd 16-vertex triangulation
+    with minimum degree 5 (in Sage's enumeration order).
+  - Conjecture 5.1's hypothesis space is *reduced* duals of such
+    triangulations: each reduced dual is obtained from the dual G' by
+    deleting one pentagonal face's 5 boundary vertices and stitching in
+    a spike vertex v_n + a merged edge (see Definition~\\ref{def:reduced-dual}).
+  - For the conjecture-5.5 phenomenon to lift to a conjecture-5.1
+    counterexample, we'd need: a reduction of C_{28} (= some choice of
+    pentagonal face F_v and rotation index i ∈ {0,…,4}) such that the
+    resulting reduced dual admits a chord-apex+Kempe colouring whose
+    two distinguished Kempe cycles K_b, K_c both have constant h_φ.
+
+This script enumerates every (F_v, i) reduction of C_{28} (and, for
+completeness, every reduction of every other 16-vertex min-deg-5
+triangulation's dual), enumerates every chord-apex+Kempe colouring of
+each reduced dual, and reports any case where h_φ is constant on
+either V(K_b) or V(K_c). The expected result (consistent with the
+empirical near-proof in Remark~\\ref{rem:heawood-empirical}) is zero
+such cases.
+
+Run with: sage experiments/check_c28_no_chord_apex_kempe_constancy.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,
+    kempe_cycle_set,
+    edge_idx,
+)
+from check_heawood_on_kempe import (
+    dual_of, heawood_numbers, vertices_of_kempe,
+)
+
+
+# Canonical graph6 of the C_{28} fullerene counterexample (from
+# experiments/verify_28_vertex_counterexample.py).
+C28_GRAPH6 = ('[kG[A?_A?_?_?K?D?@_CO?o?@_??A??@C??O??AG?C????`???a'
+              '???W???A_???F')
+
+
+def is_c28(H):
+    return H.canonical_label().graph6_string() == C28_GRAPH6
+
+
+def check_reduction(H, named, label_for_print):
+    """For one reduced dual H + named edges, enumerate every proper
+    3-edge-colouring, filter to chord-apex+Kempe colourings, and check
+    Heawood constancy on V(K_b), V(K_c), and V(K_b) ∪ V(K_c)."""
+    H.is_planar(set_embedding=True)
+    edges, colourings = proper_3_edge_colorings(H)
+    cak = [c for c in colourings if matches_chord_apex_kempe(edges, c, named)]
+    n_total = 0
+    n_const_kb = 0
+    n_const_kc = 0
+    n_const_both = 0
+    first_const = None
+    merged_idx = edge_idx(edges, named['merged'])
+    for col in cak:
+        n_total += 1
+        try:
+            h = heawood_numbers(H, edges, col)
+        except RuntimeError:
+            continue
+        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)
+        h_b = {h[v] for v in V_b}
+        h_c = {h[v] for v in V_c}
+        if len(h_b) == 1:
+            n_const_kb += 1
+        if len(h_c) == 1:
+            n_const_kc += 1
+        if len(h_b) == 1 and len(h_c) == 1:
+            n_const_both += 1
+            if first_const is None:
+                first_const = (col, V_b, V_c, h)
+    return n_total, n_const_kb, n_const_kc, n_const_both, first_const
+
+
+def iterate_reductions(G, label):
+    G.is_planar(set_embedding=True)
+    D = dual_of(G)
+    is_c = is_c28(D)
+    D.is_planar(set_embedding=True)
+    pent_faces = [f for f in D.faces() if len(f) == 5]
+    print(f"\n[{label}] triangulation order = {G.order()}, "
+          f"|V(D)|={D.order()}, |pent faces of D|={len(pent_faces)}"
+          f"{' (= C_{28})' if is_c else ''}")
+    n_red = 0
+    n_total_cak = 0
+    n_const_kb = 0
+    n_const_kc = 0
+    n_const_both = 0
+    first = None
+    for fi, face in enumerate(pent_faces):
+        for i in range(5):
+            res = apply_reduction(D, face, i, 9999)
+            if res is None:
+                continue
+            n_red += 1
+            t, ckb, ckc, both, fconst = check_reduction(
+                res['H'], res['named'], f"{label}/F{fi}/i={i}")
+            n_total_cak += t
+            n_const_kb += ckb
+            n_const_kc += ckc
+            n_const_both += both
+            if first is None and fconst is not None:
+                first = (fi, i, fconst)
+    print(f"  reductions tried        : {n_red}")
+    print(f"  chord-apex+Kempe colours: {n_total_cak}")
+    print(f"  constant on V(K_b)      : {n_const_kb}")
+    print(f"  constant on V(K_c)      : {n_const_kc}")
+    print(f"  constant on both        : {n_const_both}")
+    if first is not None:
+        fi, i, (col, V_b, V_c, h) = first
+        print(f"  *** COUNTEREXAMPLE *** in pent-face #{fi}, i={i}")
+        print(f"      colouring = {col}")
+        print(f"      V(K_b) = {V_b}, h on V(K_b) = {{h[v] for v}} = "
+              f"{{{', '.join(f'{v}:{h[v]:+d}' for v in V_b)}}}")
+        print(f"      V(K_c) = {V_c}")
+    return is_c, n_total_cak, n_const_both
+
+
+def main():
+    print("Checking reductions of all 16-vertex min-deg-5 triangulations.\n"
+          "The triangulation whose dual is the C_{28} counterexample (to "
+          "Conjecture 5.5)\nis the 3rd in Sage's enumeration order.\n")
+    start = time.time()
+    n_T = 16
+    total_T = 0
+    total_cak = 0
+    total_const = 0
+    saw_c28 = False
+    try:
+        gen = graphs.triangulations(n_T, minimum_degree=5)
+    except Exception as ex:
+        print(f"cannot enumerate triangulations({n_T}): {ex}")
+        return
+    for idx, T in enumerate(gen, start=1):
+        is_c, t, both = iterate_reductions(T, f"T#{idx}")
+        total_T += 1
+        total_cak += t
+        total_const += both
+        if is_c:
+            saw_c28 = True
+    elapsed = time.time() - start
+    print("\n" + "=" * 70)
+    print(f"Across {total_T} triangulations on {n_T} vertices "
+          f"(C_{{28}} {'YES seen' if saw_c28 else 'NOT seen'}):")
+    print(f"  total chord-apex+Kempe colourings tried : {total_cak}")
+    print(f"  total with h_φ constant on V(K_b)∧V(K_c): {total_const}")
+    print(f"[{elapsed:.1f}s]")
+    if total_const == 0:
+        print("\n✓ No chord-apex+Kempe colouring of any reduction of any "
+              "n_T = 16 triangulation\n  (including the one whose dual is "
+              "the C_{28} counterexample) has h_φ\n  simultaneously "
+              "constant on V(K_b) and V(K_c).")
+    else:
+        print(f"\n✗ {total_const} colouring(s) with full constancy found.")
+
+
+if __name__ == '__main__':
+    main()