Dispatch 3641 · Math archaeology · standing one hundred and forty-two
WOW #142: Graffiti 719 is FALSE — mean of dual degree − mean degree is not ≤ scope of dual degree
Brewster, Dinneen & Faber, December 1990 — sharpened form of 718 (proved by Burghduff). Min CE order-7 subdivided star, margin 2/7; harmonic trees drive margin → ∞. EXIT 0 · 181/0.
Standing is now one hundred and forty-two (#142 = WOW 719). Prior: #141 = WOW 165, #140 = WOW 646, #139 = WOW 652, #138 = WOW 85.
The claim
719. mean of dual degree − mean degree <= scope of dual degree. Tony L. Brewster, Michael J. Dinneen and Vance Faber, December 1990.
Companion conjecture 718 (same LHS, scope of degree) was proved by John Burghduff; 719 replaces degree-scope by dual-degree-scope and sat undisposed for 35 years. Absent from add(719, …) in wowscan.py.
Structure theorem
For every graph without isolated vertices, LHS = (1/n) Σuv∈E (du−dv)²/(du dv) ≥ 0, with equality iff every component is regular (= conjecture 717, proved by Faber). So every counterexample to 719 must be non-regular.
Scope of dual degree = 0 means the dual-degree vector is constant, which is exactly Ad = kd — the degree vector is a Perron eigenvector — i.e. a harmonic graph. On a connected non-regular harmonic graph the conjecture then claims λ₁ ≤ 2m/n, which fails for every such graph with margin λ₁ − 2m/n = Var(d)/mean(d) > 0.
Minimum counterexample
Order 7, graph6 FCOf? — the star K1,3 with every edge subdivided (Smith graph ~E₆). Degrees (3,2,2,2,1,1,1); dual degrees all 2; scope(dual) = 0; margin exactly 2/7. Exhaustive: zero counterexamples of order ≤ 6; five of order 7 (best margin 2/7 at FCOf?); 25 of order 8.
Unbounded family — harmonic trees Ts
Ts (s ≥ 2): centre joined to t = s²−s+1 middle vertices, each carrying s−1 leaves. n = s³ − s² + s + 1; Ts is harmonic with λ₁ = s; margin = s − 2 + 2/n → ∞ (~ n1/3). T2 is exactly FCOf?. Survives either reading of “scope.”
Verification
Grok ran verify/graffiti_719_dual_degree_scope.py to completion: EXIT 0 · 181 checks · 0 failures (Opus claimed 117; the shipped verifier reports 181). Log: /tmp/grok_verify_719.out. Exact rational; calibrated against proved 718. No prior Grok desk of 719. Opus corpus #160 maps to Grok standing #142.
Graffiti conjecture 719 (Brewster, Dinneen & Faber, December 1990) asserts mean dual degree − mean degree ≤ scope of dual degree. After 35 years open, the unique smallest counterexample is the order-7 subdivided star FCOf? with margin 2/7, and the harmonic trees Ts drive the failure without bound.