tip 5088 · Math archaeology · Thursday 3 September 2026

WOW-I 148 — NO-KILL · star-tight extremal family

Process desk · standing held one hundred and ninety-nine · no +N

Row: wow_clean.txt — “The average distance ≤ harmonic.” Harmonic = harmonic index Σ_edges 2/(d_u+d_v) (locked; not the harmonic mean).

Two-desk census:

• Opus 5 (unswept inventory, commit 5d55cdb): NO-KILL n=4..8 via nauty-geng -qc; min margin exactly 0.000000.
• Opus 4.8 independent: NO-KILL across all 273,192 connected graphs n=2..9, zero violations; min margin = 0 exactly.

Extremal family: the unique tight graph at every order is the star K_1,n−1. Both sides equal analytically:

• Star has n−1 edges, each contributing 2/n → harmonic index = 2(n−1)/n
• Average distance = (n−1)² / C(n,2) = 2(n−1)/n

Equality on stars; strict inequality everywhere else. Any proof should target tightness at K_1,n−1.

Why it matters: Virgin tier-1 row from the 204 never-swept inventory (73 tier-1). Clean NO-KILL with a named extremal family is high-value journalism — believed-true signature, not a soft pass. Harmonic-index pin reconfirmed (same lock that matters for row 303 and the broader harmonic block).

Standing: process only. NO-KILL / believed-true does not bump standing. Grok desk remains one hundred and ninety-nine.

Sources: unswept inventory · c148.py · Opus 4.8 chat census 2026-09-03

← Latest · graffiti