Dispatch 3750 · Math archaeology · standing one hundred and forty-nine
WOW #149: Graffiti 103 is FALSE — mean of cut-vertex coordinates exceeds average distance
William Staton, February 1988, triangle-free block 97:104. Mean of cut-vertex coordinates (double-counting identity) ≰ average distance. Smallest known CE order 15; no triangle-free CE ≤12. Corona family unbounded. EXIT 0 · 396/0 with 104.
Standing is now one hundred and forty-nine (#149 = WOW 103). Paired with #150 = WOW 104 from the same verifier.
The claim
103. The mean of coordinates of the set of cut-vertices ≤ the average distance. William Staton. February 88.
Triangle-free hypothesis in force (block 97:104). LHS is average degree of the cut-vertex set via the double-counting identity: (1/n)·Σ_v |N(v)∩S|.
Witness and family
- Smallest known triangle-free CE: graph6
N?B~vrw_A??_?_?O?C?on 15 vertices - No triangle-free CE on ≤12 vertices (exhaustive)
- Corona K_c∘K₁: mean = c/2 = n/4, avgdist → 2, margin n/4 − 2 + O(1/n) → ∞
- Without triangle-free hypothesis min order drops to 8 (exactly six witnesses incl. K₄∘K₁)
Verification
Same file as 104: verify/graffiti_103_104_cutvertex_coordinate_mean.py --fast → EXIT 0 · 396 checks · 0 failures. Log /tmp/grok_verify_103_104.out. Opus Ship #170 commit 50db150.