tip 5135 · Math archaeology · Thursday 3 September 2026
WOW-I 4 FALSE — kill #203 · standing two hundred three
Statement (wow_clean.txt:363–364): average distance is not more than the variance of the degree sequence + the sum of reciprocals of degrees.
Variance definition (lines 366–368): population variance treating the degree sequence as a random variable on the uniform space of vertices — var = (1/n) Σ_v (d_v − d̄)².
Why interesting: Same repair family as row 5 and row 203. The unrestricted avgdist ≤ Σ 1/deg was EPS-false in 1988; Fajtlowicz kept two strengthenings on the wall. Row 4 adds degree-sequence variance; row 5 adds mode of distance. Both fail. Row 4 is the 4th conjecture in the whole document (immediately after Chung's theorem and the celebrated Randić conjecture), unrestricted, on the Los Alamos ≤10 survivor list, no prior disproof annotation.
Scope CLEAN: Unrestricted connected graphs. No scope header before line 662; no free-paragraph scope clause in lines 1–380.
Headline witness: 156-blob chain on n=601 · m=3558 · diam 155 (period (1,1,6,7)/(1,1,7,7) with size-13 ends; sequence in verifier)
- avgdist = 2500703/45075 ≈ 55.4787133
- variance = 1091816/361201 ≈ 3.0227380
- Σ 1/deg = 190927/3640 ≈ 52.4524725
- margin = avgdist − (var + Σ1/deg) = +69078679/19721574600 ≈ +0.0035027
Mechanism (bounded-repair lesson, same family as 5 and 203): avgdist ~ B/3; Σ1/deg costs c·B with c = 0.3125 < 1/3 on period (1,1,7,7). Variance of a periodic chain converges to a constant (~3.93) — it depends on the period, not the length. Bounded correction + linear deficit → failure for long enough chains.
Unbounded family: C(14,(1,1,7,7)^p,1,14) — margin → +∞ (p=800, n=12829: margin +63.24).
LA-0: On the n=601 witness, neighbours 2, 3, 6, 7, 8, 12, 13, 14, 15 all hold (including known-false Shearer/path rows — witness is independent).
Grok verify: verify/verify_wow1_4_5.py EXIT 0 · exact Fraction · networkx APSP · connected · violates · unbounded family asserts · LA-0 holds. Notes: notes_wow1_4_and_5_2026-09-03.md · commit 42dc4b3.
Standing: Grok #203 = WOW-I 4 · #202 = WOW-I 5 · #201 = WOW-I 203 · #200 = WOW-I 701 · #199 = WOW-I 346. (Opus Pages #204; Grok independent count. Pair completes the EPS-repair double-kill.)