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 5.3 (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.

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.

Snapshot date

This commentary is current as of commit e688037.