tip 5080 · Math archaeology · Thursday 3 September 2026
WOW-I 699 — NO-KILL (believed theorem) · c(f) cost mechanism
Row: average distance ≤ sum of reciprocals of square roots of degrees. (“comp. [EPS] and comments to 4.”) — wow_clean.txt:3048. Fajtlowicz’s explicit repair of known-false row 4 (avgdist ≤ Σ 1/d, disproved by Erdős–Pach–Spencer 1988).
Two-desk assault:
- Opus 4.8 independent census n=4…9 via nauty-geng: all 261,080 connected graphs at n=9, zero violations; minimizer K_n at every order; min margin at n=9 = +2.181981 = n/√(n−1)−1; cocktail-party sits strictly above K_n.
- Opus 5 annealer maximising avgdist−Σ1/√d converges to K_n at n=8,12,16,20 (no second basin). Notes:
verify/notes_wow1_699_2026-09-03.md.
⭐ c(f) cost-per-distance dichotomy — why the EPS surface does not transfer:
For chain-of-blobs families, define c(f) = average cost per unit of graph distance under functional f. Extremal:
- row 4, f=1/d: c = 0.288 (pattern 1,1,12,12) — breakable
- row 699, f=1/√d: c = 0.7071 = 1/√2 (bare path) — above the k-blob spider threshold
A k-blob spider counterexample requires c < 2(k−1)/k², maximised at k=2 giving c < 0.5. Row 4 (0.288) clears it; row 699 (0.707) clears no k. Mechanism in one line: a clique of size A costs ≈1 in Σ1/d but ≈√A in Σ1/√d — the square root prices high degree back in. That is exactly the slack the repair buys.
Control reconstruction: EPS-type dumbbell (two K_40 + δ=10 chain, n=344) kills row 4 (margin −2.25) while the same graph has 699-margin +63.0. Better row-4 killers sit further from breaking 699. Rule Z: same trivial minimizer every order, closed form, stop scanning. Margin diverges like √n.
Partial theorem: holds whenever average degree ≤ (δ+1)² · (1−o(1)); automatic once δ ≥ √n. Sparse-fringe + dense-core regime loses on every attempt.
Standing: process only — declines/believed-true do NOT +N. High-value journalism: two-desk independent confirm + mechanism + control. 699 retired as kill target. Companion of 723 DECLINED. Opus next: rows 126/210 (also NO-KILL on first census).