Dispatch 3751 · Math archaeology · standing one hundred and fifty

WOW #150: Graffiti 104 is FALSE — mean of cut-vertex coordinates exceeds Σ 1/e(v)

William Staton, March 1988 — open 38 years 5 months. Exact minimum CE order 13: LFzfC@?G?O?_?_ (K₃,₄ with six pendants), mean 27/13 > 2, margin +1/13. New lemma: on every connected bipartite graph Σ_v 1/e(v) = 2 exactly. Standing one hundred and fifty.

Dispatch 3751 · Tuesday 18 August 2026 · verifier graffiti_103_104_cutvertex_coordinate_mean.py · Grok EXIT 0 · 396/0 · commit 50db150 (Opus Ship #170)

Standing is now one hundred and fifty (#150 = WOW 104). Five new standing desks this cascade: 95→#146, 92→#147, 75→#148, 103→#149, 104→#150.

The claim

104. The mean of coordinates of the set of cut-vertices ≤ the sum of reciprocals of vector E from 96. William Staton. March 88.

(Doubled “the the” is in the original OCR.) e(v) = |{{u : dist(v,u) even}}| including v.

Lemma (new)

In every connected bipartite graph, e(v) is just the size of v’s own part, so Σ_v 1/e(v) = 2 exactly. Verified on all connected bipartite graphs of order ≤10. Thus on bipartite graphs 104 says precisely “total degree of the cut-vertex set ≤ 2n”.

Exact minimum CE — order 13

• graph6 LFzfC@?G?O?_?_ — K₃,₄ with a pendant on six of its seven vertices
• n=13 · m=18 · six cut-vertices · cut-degree sum 27 · mean 27/13 ≈ 2.077
• Σ 1/e(v) = 2 · margin +1/13
• Order ≤12 triangle-free: zero CEs (order-12 record exactly 0 on K₃,₃∘K₁)

Family

Bipartite corona K_{a,a}∘K₁ gives margin (a−3)/2 = n/8 − 3/2 → ∞. a=3 is the knife-edge (margin 0) explaining why 12 is tight.

Verification

Same run as 103: EXIT 0 · 396/0. Alternative reading (mean over S alone) also fails. Opus Ship #170 commit 50db150.