face_monochromatic_pairs: empirical near-proof of Conjecture 5.1 via Lemma 5.3
Add Remark 5.5 immediately after Lemma 5.3's proof, recording the
empirical reduction of Conjecture 5.1 via the contrapositive of
Lemma 5.3: the conjecture follows from "h_phi is not constant on
V(K_b) U V(K_c)", and we have verified that non-constancy holds on
every one of 142,812 chord-apex+Kempe colourings up to n <= 20
(including the six Holton-McKay duals as a special case).
This is an independent empirical near-proof of Conjecture 5.1,
complementary to the direct (1)-(3) witness check in
Remark 5.6 / rem:conj-3-6-empirical. A structural proof of the
non-constancy claim would upgrade this to a proof of the
conjecture.
Also include two diagnostic scripts that informed the remark:
- check_shared_parity.py: parity-bucket symmetry n_{0,0} = n_{1,1},
n_{0,1} = n_{1,0} at vertices in V(K_b) cap V(K_c). 100%.
- check_cw_parity_prediction.py: structural identity
s_b XOR s_c = i_b XOR i_c XOR 1 holds at every shared vertex
(263,004 / 263,004), and the simple constancy prediction matches
exactly 50% of shared vertices per colouring with 0 perfectly
matching colourings.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
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"""For each chord-apex+Kempe colouring, walk K_b and K_c (each in
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trace order starting from the merged edge), and for every shared
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vertex v in V(K_b) cap V(K_c) record:
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i_b(v) = position of v in the K_b walk (mod 2)
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i_c(v) = position of v in the K_c walk (mod 2)
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h_phi(v)
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The proposal: under the constant-Heawood hypothesis, Lemma A forces
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each cycle's c-edge / b-edge sides to be determined by i mod 2. The
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CW order at a shared vertex v relates these. We tally the joint
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distribution of (i_b mod 2, i_c mod 2, h(v)) across all colourings
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and shared vertices, looking for a parity constraint.
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Run with: sage experiments/check_shared_parity.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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trace_kempe_cycle,
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edge_idx,
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)
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from check_heawood_on_kempe import dual_of, heawood_numbers
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def walk_positions(walk):
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"""Return dict vertex -> first-position-on-walk."""
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pos = {}
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for k, (_, leave_v) in enumerate(walk):
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if leave_v not in pos:
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pos[leave_v] = k
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return pos
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def test_one(D):
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D.is_planar(set_embedding=True)
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n_col = 0
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# Joint distribution: (i_b mod 2, i_c mod 2, h) -> count
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joint = {}
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# Per-colouring: count of shared vertices in each of the 4
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# (i_b, i_c) parity buckets, summarised.
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bucket_dist = {} # (n00, n01, n10, n11) -> count
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# Per-colouring: is sum of i_b parities over shared vertices ==
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# sum of i_c parities (mod 2)?
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sum_parity_match = 0
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sum_parity_total = 0
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# Per-colouring: is i_b(v) congruent to i_c(v) (mod 2) for ALL
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# shared vertices? Or NEVER? Or mixed?
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all_match = 0
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all_diff = 0
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mixed = 0
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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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for col in cand:
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n_col += 1
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try:
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h = heawood_numbers(H, edges, col)
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except RuntimeError:
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continue
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merged_idx = edge_idx(edges, named['merged'])
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a = col[merged_idx]
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bs = [c for c in range(3) if c != a]
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walk_b = trace_kempe_cycle(edges, col, merged_idx, (a, bs[0]))
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walk_c = trace_kempe_cycle(edges, col, merged_idx, (a, bs[1]))
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pos_b = walk_positions(walk_b)
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pos_c = walk_positions(walk_c)
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V_b = set(pos_b.keys())
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V_c = set(pos_c.keys())
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shared = V_b & V_c
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buckets = [0, 0, 0, 0] # (i_b, i_c) parities
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sum_ib = 0
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sum_ic = 0
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match_count = 0
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diff_count = 0
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for v in shared:
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pb = pos_b[v] % 2
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pc = pos_c[v] % 2
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buckets[2 * pb + pc] += 1
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sum_ib = (sum_ib + pb) % 2
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sum_ic = (sum_ic + pc) % 2
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key = (pb, pc, h[v])
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joint[key] = joint.get(key, 0) + 1
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if pb == pc: match_count += 1
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else: diff_count += 1
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if shared:
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sum_parity_total += 1
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if sum_ib == sum_ic:
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sum_parity_match += 1
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if diff_count == 0:
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all_match += 1
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elif match_count == 0:
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all_diff += 1
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else:
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mixed += 1
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bd_key = tuple(buckets)
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bucket_dist[bd_key] = bucket_dist.get(bd_key, 0) + 1
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return n_col, joint, bucket_dist, sum_parity_match, sum_parity_total, all_match, all_diff, mixed
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def main(max_n=18, time_budget_per_n=1800):
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print(f"Parity check at shared K_b cap K_c vertices, "
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f"n in [12, {max_n}]\n")
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grand_col = 0
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grand_joint = {}
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grand_bucket = {}
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grand_spm = 0; grand_spt = 0
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grand_am = 0; grand_ad = 0; grand_mix = 0
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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_col_n = 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}/{len(triangulations)}")
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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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(n_col_i, j_i, b_i, spm_i, spt_i,
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am_i, ad_i, mix_i) = test_one(D)
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n_col_n += n_col_i
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for k, v in j_i.items(): grand_joint[k] = grand_joint.get(k, 0) + v
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for k, v in b_i.items(): grand_bucket[k] = grand_bucket.get(k, 0) + v
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grand_spm += spm_i; grand_spt += spt_i
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grand_am += am_i; grand_ad += ad_i; grand_mix += mix_i
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elapsed = time.time() - start
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print(f"n={n}: {n_col_n} col., [{elapsed:.0f}s]")
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sys.stdout.flush()
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grand_col += n_col_n
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print()
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print("=" * 78)
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print(f"Grand totals (n in [12, {max_n}], {grand_col} colourings):")
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print(f"\n Joint (i_b mod 2, i_c mod 2, h_phi) distribution over "
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f"shared vertices:")
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keys = sorted(grand_joint.keys())
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total_shared = sum(grand_joint.values())
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for k in keys:
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v = grand_joint[k]
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print(f" {k}: {v} ({100*v/max(1,total_shared):.2f}%)")
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print(f"\n Per-colouring: i_b(v) == i_c(v) (mod 2) for ALL shared v?")
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print(f" all match: {grand_am}/{grand_col} "
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f"({100*grand_am/max(1,grand_col):.2f}%)")
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print(f" all differ: {grand_ad}/{grand_col} "
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f"({100*grand_ad/max(1,grand_col):.2f}%)")
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print(f" mixed: {grand_mix}/{grand_col} "
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f"({100*grand_mix/max(1,grand_col):.2f}%)")
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print(f"\n Per-colouring: sum_{{v shared}} i_b(v) ≡ sum_{{v shared}} i_c(v) (mod 2)?")
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print(f" sum-parity match: {grand_spm}/{grand_spt} "
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f"({100*grand_spm/max(1,grand_spt):.2f}%)")
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print(f"\n Most common bucket signatures (n00, n01, n10, n11):")
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bs = sorted(grand_bucket.items(), key=lambda kv: -kv[1])[:8]
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for k, v in bs:
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print(f" {k}: {v} ({100*v/max(1,grand_col):.2f}%)")
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if __name__ == '__main__':
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main()
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