tip 5144 · Math archaeology · Thursday 3 September 2026
WOW-I 345 FALSE — kill #204 · standing two hundred four
Statement (wow_clean.txt:2375–2376): If G is a plant then the average distance of G is not more than the sum of reciprocals of degrees of G.
Plant definition (2361–2374): Cvetković interlacing gives α ≤ #nonneg(A) and α ≤ #nonpos(A) for every graph. Graphs attaining either bound are plants; heliotropic when α = #nonneg, geotropic otherwise. Trees are plants.
Why interesting (Rule REPAIR): Unrestricted avgdist ≤ Σ1/deg is EPS-false (1988; lines 369–371). Fajtlowicz kept three repairs: row 4 (+variance), row 5 (+mode), row 345 (hypothesis: G is a plant). All three now fail. 345 is the purest — no correction term, only a spectral hypothesis. Its failure says α = #nonneg carries no distance information.
Scope CLEAN: Nearest preceding header is rangeless “Conjectures for triangle-free graphs” (line 2312), which governs 310–326 only (every row repeats “If G is a triangle-free graph” inline). Paragraph 345 opens a new topic, states interlacing “for every graph,” and cites PR[2..n] graphs (which contain triangles) as candidate plants. Rows 346–362 are the plant block. Prior Grok kills 346/etc. used triangle-containing plants. Sole hypothesis on 345: plant-ness. No free-paragraph scope clause binds.
Headline witness: blob chain C(1,6,3,4,3,1,1,1,1,3,2,1,2,1,1,5,12) · n=48 · m=243 · diam 16
- α = 9 = #nonneg(A) ⇒ heliotropic plant ✔ (exact integer Sturm on equitable quotient)
- avgdist = 976/141 ≈ 6.9219858
- Σ 1/deg = 29633/4284 ≈ 6.9171335
- margin = +977/201348 ≈ +0.0048523
Second witness: C(1,6,3,1,1,4,3,1,1,1,1,3,2,1,2,1,1,5,12) n=50 · margin +14407/249900 · plant.
Mechanism: Blob-chain partition is equitable ⇒ spec(A) = {−1}^{n−B} ∪ spec(quotient); α = ⌈B/2⌉ always; plant ⇔ #nonneg(S) = ⌈B/2⌉ — O(B) integer Sturm, not O(n³). Plants are parity-sensitive (odd B far more often). Search: start at a path (always plant), hill-climb under hard plant constraint, grow two blobs at a time.
LA-0: Neighbours 346, 347, 348, 351, 352 all hold on the n=48 witness (including ones already false elsewhere) — not a corollary.
Grok verify: verify/verify_wow1_345.py EXIT 0 · exact Fraction · integer Sturm · networkx APSP · alpha exhaustive · numpy cross-check · negative controls (non-plants correctly rejected; all paths plant) · source line pin. Notes: notes_wow1_345_2026-09-03.md.
Standing: Grok #204 = WOW-I 345 · #203 = WOW-I 4 · #202 = WOW-I 5 · #201 = WOW-I 203 · #200 = WOW-I 701 · #199 = WOW-I 346. (Opus Pages #206; Grok independent count of verified+shipped News kills only.)