From e5f9023d2bbd23315e6ce824566225198f7c5cbd Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 00:57:50 -0400 Subject: [PATCH] face_monochromatic_pairs: add COMMENTARY.md (proof-state summary) 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 --- papers/face_monochromatic_pairs/COMMENTARY.md | 112 ++++++++++++++++++ 1 file changed, 112 insertions(+) create mode 100644 papers/face_monochromatic_pairs/COMMENTARY.md diff --git a/papers/face_monochromatic_pairs/COMMENTARY.md b/papers/face_monochromatic_pairs/COMMENTARY.md new file mode 100644 index 0000000..de48fcf --- /dev/null +++ b/papers/face_monochromatic_pairs/COMMENTARY.md @@ -0,0 +1,112 @@ +# Empirical state of the Conjecture-5.1 proof + +This document is a snapshot of where the proof of Conjecture 5.1 +(the face-monochromatic-pair conjecture) stands after the empirical +work in `experiments/`. It is meant as commentary for the reader who +has finished reading the paper and wants to know what's been verified +computationally vs what remains to be proven structurally. + +## Summary table + +| 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) | +| 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 | +| At consecutive shared `K_b`-vertices, `i_b` parity strictly alternates | structural | ✓ universal (transition matrix has 0 within-parity transitions) | +| At odd-`i_b` shared vertices, `i_c` parity flip is forced on transition | structural | ✓ universal | + +## What's actually open + +The shortest path to a proof of Conjecture 5.1 is now: + +> Show structurally that for every chord-apex+Kempe colouring `φ` of every +> reduced dual `Ĝ'_{v,i}`, `h_φ` is not constant on `V(K_b)` (or +> equivalently on `V(K_c)`). + +This is verified on 142,812 / 142,812 colourings up to `n = 20`. It is +captured in the paper by **Corollary 5.4** (the per-cycle form of +Lemma 5.3) and **Remark 5.5** (the empirical near-proof). + +## Why the proof is harder than it looks + +Three empirical facts from the diagnostics in `experiments/` rule out +the simplest proof strategies: + +1. **The obstruction has no slack.** The minimum Heawood-flip count on + `K_b` observed across the data is **2**, attained on 12 colourings at + `n = 18`. Those colourings have *one single minority Heawood vertex* + on `V(K_b)` — flipping its sign would give constancy. So the proof + cannot rest on bulk inequalities like "at least half of `V(K_b)` has + each sign"; it must rule out every single-minority configuration. + +2. **The minority isn't anchored to a structural vertex.** For + low-flip-count colourings, the minority vertex(es) are distributed + roughly evenly across `v_n`, `A_0, …, A_4`, and "other" non-named + vertices in the rest of `G'`: + + v_n 12.86% + A_{i+1} 10.82% + A_{i+2} 8.98% + A_i 7.76% + A_{i+4} 5.31% + A_{i+3} 5.10% + other 49.18% + + So no single named vertex is *always* the minority — the proof + cannot fix on "vertex X must have Heawood opposite to the majority"; + roughly half the time the minority lives outside the named six. + +3. **No single named-vertex-pair is always a Heawood mismatch.** The + most "reliable" same-cycle pair is `(A_i, A_{i+1})` and + `(A_{i+1}, A_{i+2})` at consecutive face-boundary positions, each + with 75% mismatch rate — well short of universal. So the proof can + neither identify a specific edge that always has differing Heawood + at its endpoints, nor a specific vertex that's always minority. + +Together (1)–(3) say the obstruction is **global, not local**: there +*is* always a Heawood mismatch on `V(K_b)`, but where it sits varies by +colouring. A successful structural proof will need a global argument — +likely a topological / homological / parity-counting argument that +operates on all of `V(K_b)` simultaneously, rather than identifying a +specific forced flip. + +## Candidate mechanisms (none confirmed) + +These were explored and the corresponding diagnostics ruled them out +or revealed why they don't yield a contradiction on their own: + +- **Heawood sum identity `∑_v h_φ(v) = 0`.** Holds only ~17.6% of the + time on chord-apex+Kempe colourings; the sum can be anywhere in + `{-24, -20, …, 24}`. So this classical identity is *not* available + here. + +- **Heawood sum on a single Kempe cycle `∑_{V(K)} h_φ = 0`.** Holds only + ~23% of the time per cycle. + +- **Cycle-side balance `|L_b| = |R_b|`.** Holds only 35.43% of the + time. Constancy *would* force this exactly, but the empirical + imbalance is large in most colourings. + +- **Specific named-vertex pair always mismatches.** No such pair + exists; closest is `(A_j, A_{j+1})` at 75%. + +## Files + +- Paper text: `paper.tex`, sections 3 (Heawood number definition, + Lemma 5.2), 5 (Lemma 5.3, Corollary 5.4, Remark 5.5). +- Diagnostic scripts: see `experiments/check_heawood_*.py`, + `experiments/check_kempe_intersection_and_alternation.py`, + `experiments/check_shared_*.py`, + `experiments/check_cw_parity_prediction.py`, + `experiments/check_constancy_obstruction.py`, + `experiments/check_min_flip_structure.py`, + `experiments/check_minority_location.py`. + +## Snapshot date + +This commentary is current as of commit `e688037`.