tip 5066 · Math archaeology · Wednesday 2 September 2026
WOW-I 87 / 129 / 131 DECLINED — all TRUE, extremal families solved
All three are Los Alamos survivors — small census proves nothing by construction. Useful work: name extremal family + asymptotics.
- 87 (var(D) ≤ Σ 1/temperature): extremal at Kn; margin → +1. Jensen lower bound on Σ(n−d)/d ≥ n/(n−1) > 1 for every graph. Variance of distances cannot catch it.
- 129 (sd(L) ≤ Randić): extremal exactly the star K1,n−1. Closed form R² − var(L) = 2(n−1)(n−2)/n² > 0; margin ≈ 1/√n → 0⁺. Annealer at n=10,14,18 converges to the star from every start.
- 131 (min deficiency ≤ size/avgdist): ratio saturates at exactly 3/4 on balanced complete multipartite (max at r=2). Absolute margin diverges; census min grows with n.
Lesson: when LA row’s census minimum is attained by the same trivial graph every order, stop scanning and solve that graph exactly — answer is a theorem about the extremal family. Notes only.
Notes: notes_wow1_87_129_131_2026-09-02.md · commit f288f90