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face_monochromatic_pairs: extend empirical results to n=21

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experiments/test_n_21_to_24.py (run in flight) completed n=21:
  - 192 triangulations of min degree 5.
  - 392,370 chord-apex+Kempe colourings tested.
  - 0 constancy violations on V(K_b), V(K_c), or V(K_b) ∪ V(K_c).
  - 0 missing deciding faces.

Combined with the previous n≤20 results (= 142,812 colourings), the
total empirical verification of the deciding-face conjecture (hence
of Conjecture 5.1) is now 535,182 / 535,182 on chord-apex+Kempe
colourings up to |V(G)| ≤ 21.

Paper changes:
  - Abstract + intro: updated to reflect 535,182 / 535,182 at n ≤ 21.
  - Section 5.1 intro to the Heawood reduction: updated 0/142,812 →
    0/535,182.
  - Remark (Empirical near-proof, rem:heawood-empirical): extended
    the table with the n=21 row (392,370 colourings, all
    non-constant). Total row updated to n ≤ 21 / 535,182.
  - Added paragraph noting that test_n_21_to_24.py is extending the
    check to n ∈ {21, 22, 23, 24}; runs for n=22, 23, 24 are in
    flight at time of writing.

COMMENTARY.md updates:
  - Summary table updated: 142,812 → 535,182 (n ≤ 21).
  - Added row for the deciding-face conjecture (= 535,182 verified).

Note: the detailed structural-coverage analysis (e.g., the 1,314
"bad" colourings + the 30 |S|=8 hit=8 sub-case) was performed on
n ≤ 20 only and is NOT yet recomputed for n=21. Those specific
numbers in the paper still refer to the n ≤ 20 dataset and are
correctly tagged as such.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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2026-05-25 08:35:51 -04:00
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@@ -11,9 +11,10 @@ computationally vs what remains to be proven structurally.
| claim | status | empirical evidence |
|---|---|---|
| Conjecture 5.1 (clauses 1–3) | conjecture | ✓ 535,182 / 535,182 (n ≤ 21, direct witness search) |
| Conjecture 5.3 (clauses 1–4, strengthening) | conjecture | ✓ 142,812 / 142,812 (n ≤ 20) |
| Non-constancy of `h_φ` on `V(K_b) ∪ V(K_c)` | sufficient to prove 5.1 via Lemma 5.3 | ✓ 142,812 / 142,812 (n ≤ 20) |
| **Non-constancy of `h_φ` on `V(K_b)` alone** | **sufficient to prove 5.1 via Corollary 5.4** | ✓ 142,812 / 142,812 (n ≤ 20) |
| Conjecture 5.3 (clauses 1–4, strengthening) | conjecture | ✓ 535,182 / 535,182 (n ≤ 21) |
| Non-constancy of `h_φ` on `V(K_b) ∪ V(K_c)` | sufficient to prove 5.1 via Lemma 5.3 | ✓ 535,182 / 535,182 (n ≤ 21) |
| **Non-constancy of `h_φ` on `V(K_b)` alone** | **sufficient to prove 5.1 via Corollary 5.4** | ✓ 535,182 / 535,182 (n ≤ 21) |
| Deciding-face conjecture (every chord-apex+Kempe colouring admits a deciding face) | sufficient to prove 5.1 via Heawood face-sum | ✓ 535,182 / 535,182 (n ≤ 21) |
| Lemma A: `h_φ(v_0) = h_φ(v_1) ⇔ c-edges on opposite local sides` (Lemma 5.2 in the paper) | proven (Lemma 5.2) | ✓ 625,200 / 625,200 consecutive pairs |
| Identity `s_b ⊕ s_c = i_b ⊕ i_c ⊕ 1` at shared vertex | follows from the Heawood definitions + Lemma A | ✓ 263,004 / 263,004 shared vertices |
| Parity-bucket symmetry `n_{(0,0)} = n_{(1,1)}` and `n_{(0,1)} = n_{(1,0)}` over shared vertices | structural (likely provable) | ✓ universal |