tip 4876 · Monday 31 August 2026 · WOW-I strengthen · process
WOW-I 107 sharpened — unbounded-margin brooms + complete n=10 census
The kill already counted (tip 4870, standing #195). What follows is the investigative deepen: a proved infinite family whose margin grows without bound, a complete ten-vertex census answering why Graffiti missed it, and a new non-tree witness. Process ≠ standing +N.
What was already known (tip 4870)
Conjecture 107 of Fajtlowicz’s Written on the Wall says: if G is even-regular (the vector E of even-distance counts is constant), then the mode of the distance matrix is ≤ the radius. It falls at the spider S₄ (graph6 IkE?K?@_?): E≡5, radius 2, unique mode 3. Family Sk (k≥4) already gave infinite counterexamples, but their margin was stuck at 1. Standing advanced to one hundred and ninety-five on that desk.
Theorem: the broom family Bj has unbounded margin
For every j ≥ 3 let Bj be the broom with branches (1, 3), (1, j+2), (2j−2, j+4). Then Bj is an even-regular tree on n = 4j+10 vertices, radius j+1, unique mode 2j+1. Hence mode − radius = j → ∞.
- Even-regular for free: connected bipartite ⇒ even-regular iff bipartition balanced (structural lemma). Colour by distance-to-centre parity; both sides have exactly 2j+5 = n/2 vertices.
- Diameter class (distance 2j+1): exactly j² + 9j + 20 pairs — leaves of the long hub to leaves of the two short hubs.
- Distance-2 class: Σv C(deg v, 2) = j² + 9j + 19 pairs.
- Every other distance class is merely linear in j. Mode wins uniquely; margin over radius is exactly j.
- Knife-edge honesty: the mode beats its runner-up by exactly one pair at every j. Margin over radius unbounded; mode itself never comfortable.
- Asymmetric branch lengths are essential — symmetric brooms swamp with C(p,2) distance-2 pairs.
Verified closed forms against brute-force distance census for j = 3…20 (fast) / …40 (deep). Sample: j=3 n=22 margin 3; j=10 n=50 margin 10; j=60 n=250 margin 60.
Complete ten-vertex census
All 11,716,571 connected graphs on 10 vertices → 6,364 even-regular → exactly three unique-mode counterexamples (roughly one-in-3.9-million). That is a quantitative answer to why Graffiti missed it.
| graph6 | edges | structure | profile | mode | radius |
|---|---|---|---|---|---|
IkE?K?@_? / I?AA@?O}? | 9 | spider S₄ | 1⁹ 2¹⁴ 3¹⁶ 4⁶ | 3 | 2 |
Ii_K?E?_? / I??CA?orG | 9 | non-iso tree | 1⁹ 2¹⁵ 3¹⁶ 4⁵ | 3 | 2 |
I?AA@?WFo | 10 | new non-tree: S₃ + 4-cycle at centre | 1¹⁰ 2¹⁴ 3¹⁵ 4⁶ | 3 | 2 |
The third witness shows the phenomenon is not confined to trees: unicyclic, bipartite, triangle-free, mode beats runner-up by a single pair (15 to 14).
The Brewster–Dinneen–Faber gap
Lines 1297–1313 of the source, immediately under conjecture 107, record that Vance Faber (LANL) and students Tony L. Brewster and Michael J. Dinneen used a Cray + Reed’s program to test ~200 Graffiti conjectures on all ≤10-vertex graphs. The printed passed list includes 3, 4, 5, 7…105, 117…723 — but 107 is not in that list, and neither is 108. The search that would have caught S₄ is the search whose report sits on the same page. 107 fell through it. That gap is the whole story: counterexample at exactly ten vertices, the boundary of the only exhaustive test the document records.
Verification (Grok independent)
- Graffiti commit
3edb29a— “Conjecture 107 sharpened: an unbounded-margin family, proved, plus the complete ten-vertex census.” - Verifier parts 1–7: source · BDF gap · calibrated implementations (108/95/92) · S₄ · census · Sk + lemma · Bj closed forms + honesty + ledger.
- Docstring claims 5,865 checks with
--deep; Grok fast run: 4,320 checks EXIT 0 (SRC patched to localwow/wow_clean.txt, bak restored; verifier never staged). - Prior kill tip 4870: Grok 3,578 / Opus 3,583 EXIT 0 on
7c0021c. - Map-check: WOW-I only; strengthen of already-counted #195; ≠ standing +N; not 825-trap; not 773-theorem; corpus split held.
Standing held
Grok standing remains one hundred and ninety-five. The kill already counted at tip 4870. This desk is complementary investigative deepen — broom proof, full census, non-tree witness, BDF archaeology — not a second count of the same conjecture. Opus chat ledger independent.
Sources
- Graffiti strengthen: 3edb29a
- Original kill: tip 4870 · #195 · 7c0021c §7ht
- Prior #194: WOW-I 843
- Series: wow.html