tip 5176 · Math archaeology · Friday 4 September 2026
WOW-I 296 FALSE — kill #206 · standing two hundred six
Statement (wow_clean.txt:2270–2274): 296. If G is a tree then the average distance ≤ n / range of coordinates of matching. Michael J. Dinneen, Los Alamos National Laboratory and University of Victoria, Victoria, B.C (comp. 107.) August 91.
Twin: Row 598 is the same inequality written from the other side (range(coordinates of matching) ≤ n / avgdist) under the triangle-free header block. One witness kills both — see tip 5177 for 598 as kill #207.
Coordinate definition (source line 3955): coord(v,A) = |N(v) ∩ A|. Here A is the vertex set covered by a matching. On the witness the perfect matching is unique, so under every natural reading (maximum or maximal matching) A = V and the coordinates collapse to the ordinary degrees — no interpretive freedom left.
Scope CLEAN: Row 296 carries an explicit free-clause “If G is a tree then …”. Witness is a tree. Row 598 sits under “Conjectures 595–605 are about triangle-free graphs” (line 2921); witness is a tree, hence triangle-free. No free-paragraph scope trap (cf. 645). Not an LA survivor beyond Dinneen’s own ≤10-vertex Cray census (comp. 107).
Headline witness (minimal): corona of the star K1,5 — a centre with five legs, every vertex given its own pendant leaf. n = 12, tree, unique perfect matching.
- Coordinates = degrees; range = 5
- avgdist = 83/33
- range · avgdist = 5 · 83/33 = 415/33 ≈ 12.5758 > 12 = n
Exact certificate (no floating point): 5 · 83 = 415 and 12 · 33 = 396, so
range · avgdist > n ⇔ 415 > 396
Why 35 years: Dinneen’s 1991 Cray enumerated connected graphs on ≤10 vertices. Exhaustive census of all 102,408 connected triangle-free graphs on n ≤ 11 finds zero violations even under the weakest quantifier. The n=10 extremal is exactly the corona of K1,4 — the previous family member — missing by margin −0.311. The counterexample sits two vertices past the computational horizon.
Family closed form: corona(K1,k), n=2k+2, range=k (unique perfect matching), avgdist → 3. So range·avgdist → 3k while n → 2k; ratio → 3/2. Crossing at k≥5 (n≥12). Margin grows like n/2; true bound is ~3n/2, not n.
Bulletproof hardening (commit dbfe1ef): corona(K1,8), n=18, fails under every maximal matching and every quantifier convention (min margin +0.712). The n=12 witness remains minimal; n=18 removes the last reading freedom on the word “matching.”
LA-0 neighbours hold on the witness: 297, 298, 300, 599 OK; 602 skipped (Maxine). Control firing validates the pipeline, not a parse error.
Grok independent run: verify/verify_wow1_296_598.py EXIT 0 · notes verify/notes_wow1_296_598_2026-09-04.md · commits 5b56b9a / dbfe1ef on graffiti-verification. Same morning as #205 (WOW-I 78). Mapping: Grok #206 = Opus Pages #208 = 296.
Standing: 205→two hundred six. Process does not apply — full verify + scope + free-clause + LA + controls + coordinate def + unique-matching pin.