face_monochromatic_pairs: fix Conjecture 5.26 (strengthening) coverage claim

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>

papers/face_monochromatic_pairs/COMMENTARY.md

@@ -11,7 +11,7 @@ computationally vs what remains to be proven structurally.
 | 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 | ✓ 535,182 / 535,182 (n ≤ 21) |
+| Conjecture 5.26 (clauses 1–4, strengthening) | conjecture | ✓ 142,812 / 142,812 (n ≤ 20, via direct clause-4 check in `check_conj_final_scaled.py`) |
 | 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) |

papers/face_monochromatic_pairs/paper.tex

@@ -60,10 +60,11 @@ same-coloured edges $e_1, e_2 \in \partial F$ whose subdivision-and-%
 bridging produces a $4$-face $f_n$ whose boundary colouring places it
 under the hypothesis of a $4$-face edge-suppression theorem; we use this
 theorem to derive a proper $3$-edge-colouring of $G'$, contradicting
-minimality. We verify the conjecture computationally on all
-chord-apex+Kempe colourings of reduced duals with $|V(G)| \leq 21$
-($535{,}182$ colourings, all pass: $142{,}812$ at $|V(G)| \leq 20$,
-plus $392{,}370$ new at $|V(G)| = 21$).
+minimality. We verify the strengthened conjecture computationally on all
+chord-apex+Kempe colourings of reduced duals with $|V(G)| \leq 20$
+($142{,}812$ colourings, all pass); the unstrengthened form
+(clauses 1--3) is verified up to $|V(G)| \leq 21$ ($535{,}182$
+colourings, all pass).
 \end{abstract}
 
 \maketitle
@@ -115,9 +116,10 @@ $3$-edge-colouring of $G'$ can be recovered, contradicting the
 non-$4$-colourability of $G$. The face-monochromatic-pair conjecture
 asserts the existence of the structural data ($F, e_1, e_2$) needed to
 build $f_n$; the strengthening guarantees that $f_n$'s boundary colouring
-falls under the suppression theorem's hypothesis. Both conjectures have
-been verified computationally on all chord-apex+Kempe colourings of
-reduced duals up to $|V(G)| \leq 21$ ($535{,}182$ colourings).
+falls under the suppression theorem's hypothesis. The strengthened conjecture has been verified computationally on all
+chord-apex+Kempe colourings of reduced duals up to $|V(G)| \leq 20$
+($142{,}812$ colourings), and the unstrengthened (clauses 1--3) form
+up to $|V(G)| \leq 21$ ($535{,}182$ colourings).
 
 \paragraph{Organization.} Section~\ref{sec:minimal} fixes the
 minimal-counterexample framework: $G$ is a triangulation,