Adds experiments/test_conj_5_26_n_21_22.py, a clause-4 checker that
re-uses find_all_36_witnesses + check_clause_4 from
check_conj_final_scaled.py and runs them on n = 21, 22 with
incremental JSONL output and a 10-minute PROGRESS heartbeat.
Results (139 min wall, single thread):
n=21: 192 tri, 392,370 colourings w/ clause-1–3 witness, all pass
n=22: 651 tri, 1,786,314 colourings w/ clause-1–3 witness, all pass
total at n ≤ 22: 2,321,496 / 2,321,496 (combined with the existing
142,812 at n ≤ 20 from check_conj_final_scaled.py)
Paper edits:
- Abstract: "|V(G)| ≤ 20 (142,812)" → "|V(G)| ≤ 22 (2,321,496)" for
the strengthening; clauses-1–3 count unchanged at 535,182 / n ≤ 21.
- Intro paragraph: matching update.
- Remark rem:conj-3-8-empirical table: added n=21 and n=22 rows; new
total ($n \le 22$) = 959 triangulations, 2,321,496 colourings.
- Updated script reference in that remark to point at
check_conj_final_scaled.py + test_conj_5_26_n_21_22.py.
COMMENTARY.md summary table: Conjecture 5.26 row bumped to
2,321,496 / 2,321,496 (n ≤ 22).
Also commits the test_*_results.jsonl artifacts (with per-tri
records + n-summaries + grand summary) for reproducibility.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Previous commit (00229fa) incorrectly extended the empirical
verification of Conjecture 5.26 (strengthening, clauses 1-4) to n=21.
The running test (test_n_21_to_24.py) checks:
- Non-constancy on V(K_b), V(K_c), V(K_b) ∪ V(K_c).
- Deciding-face existence.
These verify Conjecture 5.1 (clauses 1-3) via Corollary 5.4 and via
the Heawood-face-sum route, respectively. They do NOT verify clause
(4) of the strengthening (Conjecture 5.26), which requires
constructing the subdivided graph and checking the new f_n's edge
colouring.
Conjecture 5.26 has been verified at n ≤ 20 (142,812 colourings) only,
via `check_conj_final_scaled.py` (which explicitly constructs the
clause-3 subdivision and checks clause-4). The n=21 results extend
the weaker checks but NOT the strengthening.
Paper fixes:
- Abstract: clarified that strengthened conjecture is at n ≤ 20
(142,812), unstrengthened (clauses 1-3) at n ≤ 21 (535,182).
- Intro paragraph after "we propose": same clarification.
COMMENTARY.md fix:
- Summary table: "Conjecture 5.26 (strengthening)" row reverted
to "142,812 / 142,812 (n ≤ 20)". The other rows (about Heawood-
based checks) remain at 535,182 / 535,182 (n ≤ 21).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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>
After several rounds of structural attempts and audits, the picture
has stabilized. Updated COMMENTARY.md with a "Final status" section:
- What we have proved: the Heawood-face-sum reduction (tight), the
tight structural cases (a', b', c) covering 94.97% of (G, v, i)
configurations, and the refined pigeonhole + S-cycle arguments
closing most of the residual ~5%.
- What's open structurally: full G'-pentagon fallback, and the
Kempe-cycle structural regularity "|S|=8 + hit=8 ⇒ p_G=11"
(30/30 empirical).
- Empirical closure: 100% across 142,812 chord-apex+Kempe colourings
up to |V(G)| ≤ 20.
- Why this isn't Appel-Haken in disguise: ~4 main steps + ~8
case-style sub-lemmas, vs RSST's 633. The chord-apex+Kempe
restriction does most of the work upfront.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Updated COMMENTARY.md with a "Lessons from the structural-proof
attempts" section summarizing:
- What worked: the Heawood-face-sum reduction (Theorem
deciding-face-implies-conj-5-1) and tight covering for n_k = 5
configurations.
- What didn't: n_i = 6 lemma (retracted as empirically false),
winding-number invariant (Σ = 0 under alternation, not
contradictory), case-analysis past |S| ≤ 1 (becomes discharging).
- Diagnostic: every global aggregation we've tried gives a quantity
consistent with constancy, not contradicting it. Lemma 5.2's
alternation is the right local consequence but doesn't aggregate
to a contradiction by itself.
- Open: G'-pentagon fallback structural proof; possible stronger
consequences of minimality of G beyond chord-apex.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
The user produced a concrete counterexample (whiteboard photo) showing
that h_φ can be constant on both an {a,b}-Kempe cycle K_0 and an
{a,c}-Kempe cycle K_1 sharing a colour-a edge.
Changes:
- theorem → conjecture environment, header marked **FALSE**
- New Remark records the disproof and identifies which step of the
proof attempt breaks: in the counterexample, no pair of shared
a-edges is consecutive on both cycles, so the lune-face premise
(Step 4 / Case A) doesn't apply
- Proof attempt re-tagged as "Partial proof attempt (now superseded)";
Steps 1-2 remain unconditional, Step 4 closes the sub-case where
some shared-a-edge pair is consecutive on both K_0 and K_1 (e.g.
automatically when |E(K_0) ∩ E(K_1)| = 2)
- Figure placeholder added referencing
figures/no-two-constant-kempe-counterexample.{png,pdf}
- COMMENTARY.md updated with a "Failed proof route" section so future
readers don't retread this path
Impact on Conjecture 5.1: the "Theorem 5.5 + Lemma 5.3 → 5.1" route
is closed; a structural proof of Conjecture 5.1 needs a different
angle. Lemma 5.3, Corollary 5.4, and the 142,812/142,812 empirical
near-proof all stand.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Standalone commentary document for readers of the paper:
- Headline table mapping each empirical / structural claim to its
proof status and the verification numbers we have.
- Statement of "what's actually open": the structural proof of
non-constancy of h_phi on V(K_b) (alone), which reduces to
Conjecture 5.1 via Corollary 5.4.
- Three reasons the proof appears to be hard:
(1) the obstruction has no slack (min flip count 2 -> 1 minority
vertex);
(2) the minority is not anchored to a structural vertex (~half
live on "other" non-named vertices);
(3) no single named-vertex-pair is always a mismatch (max 75%).
- List of candidate mechanisms ruled out by diagnostics:
- global sum identity, per-cycle sum identity,
- cycle-side balance |L| == |R|,
- specific-pair-always-mismatches.
- Index of diagnostic scripts in experiments/.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis