face_monochromatic_pairs: verify G'-pentagon fallback empirically on bad colourings
Three verification scripts: experiments/check_30_residual.py and check_30_residual_v2.py: attempt to identify the hypothesized residual case (|S| = 8 AND p_hit = p_total = 8) where all G'-pentagons would be hit by S forcing the fallback to require G'-heptagons. Result: 0 such colourings — the conditional doesn't occur empirically. experiments/check_gprime_pentagon_always_works.py: direct check that across all 1,314 bad colourings, at least one G'-pentagon has its boundary entirely in V(K_b) ∪ V(K_c). RESULT: 1,314 / 1,314 = 100.00% have an uncovered G'-pentagon. So the G'-pentagon fallback conjecture (Conjecture gprime-pentagon-fallback) is empirically true on ALL chord-apex+ Kempe colourings — both the "tight" ones (handled structurally by Theorem deciding-face-partial-extended) and the "bad" ones (where Lemma flank-covering-hex fails). Implication: the residual cases I worried about (where the fallback would need to be relaxed to length ≢ 0 mod 3) DO NOT OCCUR. So the Conjecture (G'-pentagon fallback) suffices to close the deciding- face conjecture in full empirical generality. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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"""Simpler version: enumerate the residual (|S|=8, hit=p_total=8)
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colourings without requiring v_parent identification. For each,
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determine:
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- the n_k sequence of the reduction (from the reduced dual's
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F_k structure),
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- whether any G'-face (length ≢ 0 mod 3) is uncovered.
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The earlier check_S_face_structure.py showed at most 30 |S|=8 cases
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with hit = 8, but didn't constrain p_total. Of those, ≤30 have
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p_total = 8 (= the residual).
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Run with: sage experiments/check_30_residual_v2.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 is_g_prime_face(f, named):
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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):
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D.is_planar(set_embedding=True)
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residual_colourings = []
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other_bad = []
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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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# Compute the n_k sequence for this reduction
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# n_k = degree of the face F_k of original G' = length of
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# the corresponding face in the reduced dual after subtracting
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# the reduction effect.
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# Simpler: the flank face F^♭_{i, i+1} has length n_i - 1.
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# So n_i = length of F^♭_{i,i+1} + 1.
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# Get all faces:
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named_face_lengths = {}
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for f in H.faces():
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fset = {frozenset(e) for e in f}
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if named['side_0'] in fset and named['spike'] in fset:
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named_face_lengths['flank_lower'] = len(f)
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if named['spike'] in fset and named['side_1'] in fset:
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named_face_lengths['flank_upper'] = len(f)
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if (named['side_0'] in fset and named['side_1'] in fset
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and named['merged'] in fset):
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named_face_lengths['outer'] = len(f)
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if (named['merged'] in fset and named['side_0'] not in fset
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and named['side_1'] not in fset
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and named['spike'] not in fset):
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named_face_lengths['merged'] = len(f)
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n_i = named_face_lengths.get('flank_lower', 0) + 1 # = n_i, where the flank covers
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n_ip1 = named_face_lengths.get('flank_upper', 0) + 1
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# n_{i+3} = F_merged length + 2
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n_ip3 = named_face_lengths.get('merged', 0) + 2
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# n_{i+2} + n_{i+4} from outer
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outer_len = named_face_lengths.get('outer', 0)
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# outer_len = n_{i+2} + n_{i+4} - 3
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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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# bad colouring
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# Count G'-pentagons total and hit
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p_total = 0
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p_hit = 0
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non_pent_uncovered = []
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for f in H.faces():
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if not is_g_prime_face(f, named): continue
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L = len(f)
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verts = {u for (u, v) in f} | {v for (u, v) in f}
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if L == 5:
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p_total += 1
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if verts & S: p_hit += 1
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else:
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if L % 3 != 0 and verts.issubset(V_union):
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non_pent_uncovered.append(L)
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# Is this a "residual" case?
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S_size = len(S)
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if S_size == 8 and p_total == p_hit:
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residual_colourings.append({
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'n_i': n_i, 'n_ip1': n_ip1, 'n_ip3': n_ip3,
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'outer_len': outer_len,
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'p_total': p_total, 'p_hit': p_hit,
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'S_size': S_size,
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'non_pent_uncovered': non_pent_uncovered,
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})
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elif S_size == 8 and p_hit == p_total - 1 and p_total >= 7:
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# Border case: only 1 pentagon uncovered
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pass
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return residual_colourings
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def main(max_n=20, time_budget_per_n=1800):
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print("Detailed analysis of |S|=8, p_hit = p_total residual "
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"chord-apex+Kempe colourings.\n")
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grand_residuals = []
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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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resids = test_one(D)
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for r in resids:
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r['n_G'] = n
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r['tri_idx'] = tri_idx
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n_count += len(resids)
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grand_residuals.extend(resids)
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elapsed = time.time() - start
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print(f"n={n}: {n_count} residual colourings [{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 residual colourings: {len(grand_residuals)}")
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if grand_residuals:
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# n_k sequence distribution
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seq_dist = {}
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for r in grand_residuals:
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key = (r['n_i'], r['n_ip1'], r['n_ip3'], r['outer_len'])
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seq_dist[key] = seq_dist.get(key, 0) + 1
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print("\n(n_i, n_{i+1}, n_{i+3}, F_outer length) distribution:")
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for k, c in sorted(seq_dist.items(), key=lambda x: -x[1]):
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print(f" {k}: {c}")
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# For each, does a non-pentagon G'-face provide a deciding face?
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has_non_pent = sum(1 for r in grand_residuals if r['non_pent_uncovered'])
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print(f"\nResidual colourings with at least one length-≢0-mod-3 "
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f"G'-face uncovered: {has_non_pent} / {len(grand_residuals)} "
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f"({100*has_non_pent/len(grand_residuals):.2f}%)")
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# Lengths of those non-pent G'-faces
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len_dist = {}
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for r in grand_residuals:
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for L in r['non_pent_uncovered']:
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len_dist[L] = len_dist.get(L, 0) + 1
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print("Lengths of uncovered non-pentagon G'-faces:")
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for L, c in sorted(len_dist.items()):
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print(f" |f| = {L}: {c}")
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if __name__ == '__main__':
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main()
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