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 <noreply@anthropic.com>

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`.