face_monochromatic_pairs: stronger structural regularity at |S|=8
experiments/check_S8_hit8_pG.py finds that all 30 |S|=8 bad
colourings with hit = 8 have p_G = 11 EXACTLY. Not p_G ∈ {9, 10, 11}
as I'd expected, but always 11.
This means: when |S| = 8 and 8 G'-pentagons are hit, the parent
triangulation v has NO degree-5 neighbours (= all 5 neighbours have
degree ≥ 6), and hence the reduced dual has 12 - 1 - 0 = 11
G'-pentagons. Three G'-pentagons are uncovered, not merely one.
Updated Remark (gprime-pigeonhole-stop) in paper to reflect this
stronger regularity: the size of S = V \ (V(K_b) ∪ V(K_c)) is
structurally tied to the count of pentagonal F_k adjacent to F_v in
chord-apex+Kempe colourings. A non-empirical proof of this is open.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
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"""For |S| = 8 bad colourings with hit = 8 G'-pentagons,
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report the distribution of p_G (# total G'-pentagons in the reduced
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dual). The structural claim is that p_G ≥ 9 always in this sub-case.
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If p_G ≥ 9 always: hit = 8 < p_G ≥ 9, so ≥ 1 G'-pentagon uncovered. ✓
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If p_G = 8 sometimes: those would be true structural gaps.
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Run with: sage experiments/check_S8_hit8_pG.py
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"""
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import os
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import sys
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import time
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from sage.all import Graph
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from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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kempe_cycle_set,
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edge_idx,
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)
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from check_heawood_on_kempe import dual_of, vertices_of_kempe
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def is_g_prime_pentagon(f, named):
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if len(f) != 5: return False
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fset = {frozenset(e) for e in f}
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return not (named['side_0'] in fset or named['side_1'] in fset
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or named['spike'] in fset or named['merged'] in fset)
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def test_one(D, n_G, tri_idx):
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D.is_planar(set_embedding=True)
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s8_hit8_cases = [] # list of (p_G, parent_v_cyc_degs)
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# Get cyclic degree sequence of triangulation vertices for the
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# face we're examining
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for face in D.faces():
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if len(face) != 5: continue
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for i_red in range(5):
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res = apply_reduction(D, face, i_red, 9999)
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if res is None: continue
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H = res['H']; named = res['named']
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H.is_planar(set_embedding=True)
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edges, colorings = proper_3_edge_colorings(H)
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cand = [c for c in colorings
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if matches_chord_apex_kempe(edges, c, named)]
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v_n = 9999
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for col in cand:
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target = {named['side_0'], named['spike']}
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lower_flank = None
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for f in H.faces():
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if target.issubset({frozenset(e) for e in f}):
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lower_flank = f; break
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if lower_flank is None or len(lower_flank) != 5: continue
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arc_verts = [e[0] for e in lower_flank]
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if v_n not in arc_verts: continue
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k = arc_verts.index(v_n)
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cyc = arc_verts[k:] + arc_verts[:k]
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A_i = next(iter(named['side_0'] - {v_n}))
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A_ip1 = next(iter(named['spike'] - {v_n}))
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if cyc[1] == A_i and cyc[4] == A_ip1:
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P_1, P_2 = cyc[2], cyc[3]
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elif cyc[1] == A_ip1 and cyc[4] == A_i:
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P_2, P_1 = cyc[2], cyc[3]
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else: continue
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merged_idx = edge_idx(edges, named['merged'])
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c_col = col[merged_idx]
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c_0_col = col[edge_idx(edges, named['side_0'])]
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c_1_col = col[edge_idx(edges, named['side_1'])]
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e_AiP1 = edge_idx(edges, frozenset((A_i, P_1)))
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e_P1P2 = edge_idx(edges, frozenset((P_1, P_2)))
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if e_AiP1 is None or e_P1P2 is None: continue
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if col[e_AiP1] != c_1_col or col[e_P1P2] != c_0_col:
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continue
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a = c_col
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other = [x for x in range(3) if x != a]
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kc_b = kempe_cycle_set(edges, col, merged_idx, (a, other[0]))
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kc_c = kempe_cycle_set(edges, col, merged_idx, (a, other[1]))
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V_b = vertices_of_kempe(edges, kc_b)
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V_c = vertices_of_kempe(edges, kc_c)
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V_union = V_b | V_c
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S = set(H.vertices()) - V_union
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if P_1 in V_union: continue
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if len(S) != 8: continue
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p_total = 0
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p_hit = 0
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for f in H.faces():
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if not is_g_prime_pentagon(f, named): continue
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p_total += 1
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verts = {u for (u, v) in f} | {v for (u, v) in f}
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if verts & S: p_hit += 1
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if p_hit == 8:
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s8_hit8_cases.append({
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'p_G': p_total, 'n_G': n_G, 'tri_idx': tri_idx,
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'i_red': i_red,
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})
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return s8_hit8_cases
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def main(max_n=20, time_budget_per_n=1800):
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print("|S| = 8 colourings with hit = 8: p_G distribution\n")
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grand_cases = []
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for n in range(12, max_n + 1):
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start = time.time()
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try:
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triangulations = list(graphs.triangulations(n, minimum_degree=5))
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except Exception as ex:
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print(f"n={n}: cannot enumerate ({ex})")
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continue
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n_count = 0
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for tri_idx, G in enumerate(triangulations):
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if time.time() - start > time_budget_per_n:
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print(f" n={n}: timeout at tri {tri_idx}")
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break
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G.is_planar(set_embedding=True)
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D = dual_of(G)
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cases = test_one(D, n, tri_idx)
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grand_cases.extend(cases)
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n_count += len(cases)
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elapsed = time.time() - start
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print(f"n={n}: {n_count} (|S|=8, hit=8) cases [{elapsed:.0f}s]")
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sys.stdout.flush()
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print()
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print("=" * 70)
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print(f"Total |S|=8 hit=8 cases: {len(grand_cases)}")
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p_G_dist = {}
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for c in grand_cases:
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p_G_dist[c['p_G']] = p_G_dist.get(c['p_G'], 0) + 1
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print("\np_G distribution among |S|=8 hit=8 cases:")
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for p in sorted(p_G_dist):
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print(f" p_G = {p}: {p_G_dist[p]}")
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if 8 in p_G_dist and p_G_dist[8] > 0:
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print("\n⚠ STRUCTURAL GAP: p_G = 8 occurs with hit = 8")
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print(" So the G'-pentagon fallback is genuinely contradicted")
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print(" in these cases.")
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else:
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print("\n✓ p_G = 8 NEVER co-occurs with hit = 8. So in |S| = 8")
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print(" bad colourings, p_G ≥ 9 whenever hit = 8, giving")
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print(" ≥ 1 uncovered G'-pentagon. The fallback is empirically")
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print(" closed.")
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if __name__ == '__main__':
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main()
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Binary file not shown.
@@ -1347,14 +1347,20 @@ $8$ & $252$ & $8$ & $\boldsymbol{9}$\,$^\dag$ & $1$ \\
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$10$ & $36$ & $7$ & $8$ & $1$ \\
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\end{tabular}
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\end{center}
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$^\dag$ Empirically, the combination $|S| = 8$ with \# pent. hit $= 8$
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\emph{and} $p_G = 8$ never occurs --- so although both $\text{hit} = 8$
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and $p_G = 8$ are individually possible at $|S| = 8$, they are never
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simultaneous. (\texttt{experiments/check\_30\_residual\_v2.py}.)
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This is a structural fact about chord-apex+Kempe colourings that we
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do not have a non-empirical proof of, but its empirical confirmation
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shows the G'-pentagon fallback closes the $|S| = 8$ case along with
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the other $|S|$ values.
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$^\dag$ Empirically, every chord-apex+Kempe colouring with $|S| = 8$
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and \# pent. hit $= 8$ has $p_G = 11$ exactly --- not merely $p_G \ge 9$.
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That is, every one of the $30$ cases that achieves hit $= 8$ at $|S| = 8$
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has all $5$ of $v$'s neighbours in the parent triangulation of
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degree $\ge 6$, so no $F_k$ adjacent to $F_v$ is pentagonal, leaving
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$p_G = 12 - 1 - 0 = 11$ G'-pentagons in the reduced dual. So when
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hit $= 8$, $3$ G'-pentagons are uncovered ---
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\emph{not merely~$1$}. (\texttt{experiments/check\_S8\_hit8\_pG.py}.)
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This is a much stronger structural regularity than we expected,
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suggesting that in chord-apex+Kempe colourings, the size of
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$S = V \setminus (V(K_b) \cup V(K_c))$ is structurally tied to
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the count of pentagonal $F_k$ adjacent to $F_v$ in a way that always
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leaves G'-pentagons uncovered. A structural (non-empirical) proof of
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this regularity is open.
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Combining Theorem~\ref{thm:deciding-face-partial-extended} with the
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empirical structural facts above, the $G'$-pentagon fallback
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