math-research/papers/face_monochromatic_pairs/COMMENTARY.md

# 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 5.26 (clauses 1–4, strengthening) | conjecture | ✓ 2,321,496 / 2,321,496 (n ≤ 22, via direct clause-4 check in `check_conj_final_scaled.py` for n ≤ 20 + `test_conj_5_26_n_21_22.py` for n ∈ {21, 22}) |
| 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 |
| 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`.

## Failed proof route via edge-sharing Kempe constancy

A natural-looking strategy to prove Conjecture 5.1 was:

> **Conjecture (now disproved):** If $K_0$ is an $\{a,b\}$-Kempe cycle
> of $\varphi$ and $K_1$ is an $\{a,c\}$-Kempe cycle of $\varphi$ that
> shares an edge with $K_0$, then $h_\varphi$ cannot be constant on
> both $V(K_0)$ and $V(K_1)$ simultaneously.

Combined with Lemma 5.3 (no clause-3 witness $\Rightarrow$ constancy
on both $V(K_b)$ and $V(K_c)$), this would have closed Conjecture 5.1.
It is **false** by a concrete counterexample (Figure in `paper.tex` at
`\ref{fig:no-two-constant-kempe-counterexample}`). A partial proof
attempt is preserved in the paper alongside the disproof:

- Step 1 (local CW structure) — unconditional.
- Step 2 (forced odd-crossing $\Rightarrow |E(K_0) \cap E(K_1)|$ even
  and $\geq 2$) — unconditional.
- Step 3 (Heawood face-sum mod 3) — unconditional but does not yield
  a contradiction on its own.
- Step 4 (lune-face Case A) — closes the sub-case where two shared
  a-edges are consecutive on \emph{both} cycles (automatic when
  $|E(K_0) \cap E(K_1)| = 2$).
- Step 5 (general case) — open / now known false via the
  counterexample.

The counterexample shows the general case of the conjecture is
unsalvageable; the search for a structural proof of Conjecture 5.1
will need a different angle.

## Lessons from the structural-proof attempts (commit `fd4b89a` onward)

After working through three structural-proof routes to the deciding-face
conjecture and beyond, here's what's known and what's still open:

### What worked
- **Reduction** (Theorem `thm:deciding-face-implies-conj-5-1`): Conjecture 5.1
  follows from the existence of a deciding face — a face of the reduced
  dual with boundary ⊆ V(K_b) ∪ V(K_c) and length not divisible by 3 —
  via Heawood's classical face-sum identity. This is clean and tight.
- **Tight covering for $n_k = 5$** (Lemmas `lem:flank-covering-base` and
  `lem:outer-face-covering-base`): if any of (i) $n_i = 5$, (ii) $n_{i+1} = 5$,
  or (iii) $n_{i+2} = n_{i+4} = 5$ holds, the flank or outer face is a
  tight structural deciding face. Covers 94.97% of the 7,930 empirical
  (G, v, i) configurations up to $|V(G)| \le 20$.
- **Partial pigeonhole for the G'-pentagon fallback** (Lemma
  `lem:gprime-pigeonhole`): if at most one vertex is uncovered by
  V(K_b) ∪ V(K_c), at least one G'-pentagon is a deciding face. Combined
  with the tight cases, covers ~91% structurally.

### What didn't work
- **n_i = 6 flank-covering lemma** — *the lemma is empirically false*.
  9,228 / 142,812 chord-apex+Kempe colourings hit sub-case (ii.B) of
  Case (b), and 1,314 of those have P_1 ∉ V(K_b) ∪ V(K_c), falsifying
  the lemma. Retracted in commit `873c2cc`.
- **Winding-number / topological invariant** (commit `fd4b89a`).
  Σ-of-turn-signs around K_b under Lemma 5.2 alternation = 0. This is
  *consistent* with K_b being a simple closed planar curve bounding a
  non-empty region — not a contradiction. The natural topological
  invariant doesn't distinguish chord-apex+Kempe colourings under
  constancy from generic K_b's.
- **Closing the case analysis past |S| ≤ 1** — would require finer
  graph-structural input, fragmenting the case count. This is exactly
  the discharging flavour of the traditional reducible-configurations
  approach (Appel-Haken / RSST / Gonthier) to the Four Colour Theorem
  itself; we stopped to avoid replaying that route in a different
  vocabulary.

### Diagnostic observation
Across all attempts, Lemma 5.2's alternation (= constancy on V(K_b)
implies c_1-edges alternate sides along K_b) is the natural local
consequence of constancy, but each global aggregation we've tried
produces a quantity that's *consistent* with constancy rather than
contradicting it:

- Heawood face-sum mod 3 on a specific face: requires identifying *which*
  face — that's case analysis.
- Winding number of K_b: zero under alternation, which only says K_b
  isn't a face boundary (true, not contradictory).
- Identity s_b ⊕ s_c = i_b ⊕ i_c ⊕ 1: reduces under constancy to
  c_b ⊕ c_c = 1, true but not contradictory.

This suggests the contradiction in the chord-apex+Kempe setting is
*not* captured by local-or-aggregable invariants of K_b alone.

### What's left open

- **Conjecture (Deciding face)**: 100% empirically verified
  (142,812 / 142,812), 94.97% structurally proven, ~5% reducible to
  Conjecture (G'-pentagon fallback), itself 100% empirical but
  structurally open.
- **G'-pentagon fallback structural proof beyond |S| ≤ 1**: needs a
  Kempe-cycle structural result describing *which* G'-pentagons the
  uncovered vertices of chord-apex+Kempe colourings can hit.
- **Minimality of G**: invoked to derive chord-apex (= colour equality
  spike = merged) then forgotten. Possibly minimality has a stronger
  consequence we haven't extracted.

## Final status (commit `2d8c679`)

### What we have proved

1. **Theorem (Conjecture 5.1 ⇐ Deciding face)**: clean reduction via Heawood's
   classical face-sum identity. Tight.

2. **Tight structural cases for the Deciding face conjecture**:
   - (a') $n_i = 5$ → flank face $F^\flat_{i, i+1}$ of length 4.
   - (b') $n_{i+1} = 5$ → flank face $F^\flat_{i+1, i+2}$.
   - (c)  $n_{i+2} = n_{i+4} = 5$ → outer face $F^\flat_{\text{outer}}$ of length 7.

   Empirical coverage: 7,531 / 7,930 (G, v, i) configurations = **94.97%**.

3. **Refined pigeonhole + S-cycle structure**: for bad colourings
   (= where Lemma flank-covering-hex fails), the uncovered vertex set
   $S$ is even, forms a 2-regular subgraph (= a single cycle), and is
   bounded so that # G'-pentagons hit by S < $p_G$. This closes most
   bad cases structurally.

4. **Empirical closure**: the G'-pentagon fallback conjecture is true on
   1,314 / 1,314 bad colourings, hence on all **142,812 / 142,812**
   chord-apex+Kempe colourings up to $|V(G)| \leq 20$.

### What's open structurally

1. **G'-pentagon fallback in full generality** — proven via pigeonhole for
   $|S| \leq 1$, sketched for higher $|S|$ via empirical bounds (max hit,
   min $p_G$), but no fully structural proof of the empirical bounds.

2. **The "|S| = 8 + hit = 8 ⇒ $p_G = 11$" regularity** — striking
   empirical fact (30/30 cases) but no Kempe-cycle structural argument
   for it. The marginal $|S|$ ↔ # pent $F_k$ coupling doesn't hold;
   the regularity is a joint structural property of specific
   $(|S|, \text{hit})$ pairs.

### What the path is

The structural proof has stabilized at the following form:

```
Conjecture 5.1 (face-monochromatic-pair)
  ⇐ Theorem (Conj 5.1 ⇐ Deciding face)     [tight]
Conjecture (Deciding face)
  ⇐ Theorem (Extended partial proof)       [tight on 7,531/7,930]
  ⇐ Conjecture (G'-pentagon fallback)       [empirical 1,314/1,314]
  ⇐ Lemma (G'-pentagon pigeonhole, |S| ≤ 1) [tight on |S| ≤ 1 ≈ 91%]
  + open: structural |S|-cycle arguments for |S| ≥ 2.
```

So Conjecture 5.1 is reduced to **two open structural conjectures**:

  (i) The G'-pentagon fallback conjecture (empirically 100%).
  (ii) Kempe-cycle-structural lemmas about chord-apex+Kempe colourings
       sufficient to close the |S| ≥ 2 cases of the fallback.

Closing (i) and (ii) structurally would give a full structural proof.
The empirical 100% across 142,812 colourings is strong evidence both
are true.

### Why this isn't Appel-Haken in disguise

The structural proof has 4 main steps + ~8 case-style sub-lemmas (vs
RSST's 633 cases). The compression comes from the chord-apex+Kempe
restriction doing most of the heavy lifting upfront — specifically,
Lemma 5.3 (no-witness ⇒ constancy on V(K_b) ∪ V(K_c)) and Lemma 5.X
(Kempe-spike, forced edge containments) pre-restrict the problem to
a very constrained colour configuration. The Heawood face-sum
identity then converts the local constancy into a mod-3 obstruction,
and the case analysis closes specific structural sub-buckets.

The remaining open conjectures are at the chord-apex+Kempe class
level, not at the level of arbitrary minimal counterexamples to 4CT.

### Snapshot date

This commentary is current as of commit `2d8c679` (2026-05-25).