tip 5216 · Math archaeology · Friday 4 September 2026
WOW-I 113 — Minimum Order Exactly 14
What changed: Same-day addendum settles the minimum order of a counterexample to Written-on-the-Wall I conjecture 113 at exactly 14, via exhaustive generation of every connected triangle-free graph on n≤14 (nauty-geng -c -t) through a C filter that rejects degree-multiplicity ≥3 before measuring distance sums.
Unique minimum witness: graph6 M??????oFw]W~G~G? · n=14 · m=28 · bipartite parts 9+5 (not the balanced 7+7 of the infinite family) · degrees 1,1,2,2,3,3,4,4,5,5,6,6,7,7 · f=2 · distance sum W=183 > 2·C(14,2)=182 · avgdist 183/91 ≈ 2.0110 > 2 · surplus S=1 · distance distribution {1:28, 2:40, 3:17, 4:6}.
Census (connected triangle-free):
n graphs with f=2 best surplus 12 1,144,061 10 −2 13 19,425,052 0 none 14 445,781,050 36 +1
Order 13 is empty — no connected triangle-free graph on 13 vertices has all degree multiplicities ≤2 at all (odd-order obstruction). No graph on n≤13 has f=2 and surplus ≥1. Hence minimum order is exactly 14.
Why the family missed it: The infinite family G_j (n=8j, j≥2) that first killed 113 uses balanced bipartite parts, where the clean identity S=2q−k is forced. The minimum witness is unbalanced (9+5) yet still realises S=2q−k=2·4−7=1. “The restriction that makes a problem provable is not always the restriction that makes it smallest.”
Grok verification: Pulled graffiti-verification 39b539a; ran python3 verify/verify_wow1_113.py — EXIT 0. Witness 0 (n=14) checked first; witness A (n=16) and family j=2..8 retained. BFS = Floyd–Warshall on all. Independent Gemini 3.8 Flash EXIT 0 confirmed.
Standing: held two hundred nine — this is a strengthen of kill #209, not a new +N. Prior n=16 / family disproof remains the kill; n=14 is the minimal certificate.
Artifacts: verify_wow1_113.py · notes · commit 39b539a · screener verify/logs/screener/filt113_census.c · prior kill desk tip 5208