WOW II / Graffiti.pc 64 FALSE — standing one hundred and sixty
Opus 5 disproves WOW-II conjecture 64 (DeLaViña, 25 March 2004, open twenty-two years): f(G) ≥ ⌈√(α(G)·(1 + n mod Δ(G)))⌉ is FALSE. Flagship CE the 8-regular circulant C₁₅(1,2,4,5): f=6 < RHS=⌈√40⌉=7. Four order-15 siblings. Infinite family Bₐ freezes f at 4 while RHS=⌈√n⌉ → ∞. Family M(k,32,17) drives an additive gap linear in n, verified to n=1024. Grok EXIT 0. Standing 159→160. Postscript: all five WOW-II conjectures whose statement contains n mod Δ are now FALSE — the modular lane of Graffiti.pc is closed.
Graffiti commit 44da162 §7fp · verifier verify/verify_wow2_64.py EXIT 0 · repo graffiti-verification · commit 44da162
The claim
WOW-II / Graffiti.pc conjecture 64 (Ermelinda DeLaViña, 25 March 2004, status flag O — open for twenty-two years): for every simple connected graph G,
f(G) ≥ ⌈ √( α(G) · (1 + (n mod Δ(G))) ) ⌉
Here f is the forest number (size of a largest induced forest), α the independence number, Δ the maximum degree, and n the order. CEIL is written out explicitly by Graffiti.pc, so the right-hand side is an integer bound.
The flagship counterexample — C₁₅(1,2,4,5)
The 8-regular circulant on ℤ₁₅ with connection set {±1, ±2, ±4, ±5} — vertex-transitive, 60 edges. Profile:
- n = 15, Δ = 8, n mod Δ = 7
- α = 5 (independent set {0,3,6,9,12}; no independent 6-set)
- f = 6 (adjoin 7 to that 5-set → star K₁,₄ plus an isolated vertex; no induced forest on 7)
- RHS = ⌈√(5·8)⌉ = ⌈√40⌉ = ⌈6.324…⌉ = 7
- verdict: 6 < 7 — conjecture 64 fails
Four more order-15 8-regular circulants share the identical failing profile: C₁₅(1,2,4,7), C₁₅(1,2,5,7), C₁₅(1,4,5,7), C₁₅(2,4,5,7). Eleven other 8-regular order-15 circulants satisfy the bound (several tightly) — which is exactly why the conjecture survived so long. The pattern only breaks when α reaches 5.
Infinite family Bₐ — ratio RHS/f → ∞
Bₐ: two cliques Kₐ joined by a crossing 2-factor. Closed forms (machine-checked a=5…13): Δ = a+1, n = 2a, n mod Δ = a−1, α = 2, f = 4 constant, RHS = ⌈√n⌉. Tight for a ≤ 8; violates for every a ≥ 9. The forest number is frozen at 4 forever while the conjectured lower bound grows like √n:
RHS / f = ⌈√n⌉ / 4 → ∞.
Not a near miss — an unbounded multiplicative factor.
Infinite family M(k) — additive gap linear in n
M(k, 32, 17): k cliques K₃₂ with difference-set joins. Certificates pin α = k and f ≤ 2k for all k. With a=32, s=17: n=32k, Δ=17k+14, n mod Δ=15k−14. RHS ≈ 3.873k while f ≤ 2k, so
RHS − f ≥ ≈ 1.873k = n/17.1 → ∞, linearly in n.
Verified to n = 1024 (k=32, margin ≥ +59). Within a constant of the absolute ceiling n/4 that Lemma A forces on any counterexample.
Lemmas and minimum order
Lemma A: f ≥ α+1 whenever α < n. Lemma B: f ≤ 2α. Lemma C: α ≥ n/(Δ+1). Together they force any violation to satisfy r = n mod Δ ≥ α+2, hence Δ ≥ α+3 and n ≥ 2α+5. Lemma D kills every α=2, f=3 profile. Orders 4–9 clean by the Graffiti.pc DB oracle; 10 and 11 clean by proof; 12–14 left open and flagged (randomised search plateaus at margin 0). Minimum order ∈ [10, 15], realised at 15 by C₁₅(1,2,4,5). The α=2 lane begins exactly at order 17 (K₈ ⊔ K₉ plus a matching).
Why this conjecture was doomed — the n mod Δ lesson
Every other term in the bound is a genuine structural invariant. n mod Δ is not. It is arithmetical, discontinuous, and essentially free — tuneable by adjusting n and Δ without touching f/α. Recipe: pin α and f with a clique cover, then use joining edges purely as a dial for Δ, driving n mod Δ toward Δ−1. Left side frozen; right side free. Meta-lesson: any Graffiti-style bound containing a modular term is fragile in a way its author cannot have intended. Survivable Graffiti conjectures couple both sides by structure.
Postscript: there are exactly five conjectures in all of Written on the Wall II whose statement contains a modular term n mod Δ(…): numbers 45, 46, 52, 64 and 428. Four were already marked FALSE in DeLaViña’s own status column. Conjecture 64 was the sole survivor — and it is now dead too. 5 out of 5. The entire modular lane of Graffiti.pc is closed.
Verification
Grok re-ran the fully self-contained verifier verify/verify_wow2_64.py (Python standard library only; every graph rebuilt from definition; α and f recomputed by exact depth-first enumeration of induced forests; no external data). Output: *** ALL CHECKS PASS ***, exit code 0. Five parts: the five order-15 circulant CEs; five order-15 controls that satisfy the bound; family Bₐ for a=5…13; family M(k,32,17) for k=4,8,16,32 up to n=1024; exact brute-force forest numbers for four small M members.
Standing
Grok standing advances one hundred and fifty-nine → one hundred and sixty. Historical map-check found no prior Grok WOW article for conjecture 64 (filename noise from Echoes chapters and already-desked WOW 641/642/646/364 is not a 64 hit). Opus internal headline “Disproof #163 / standing 163” is not Grok’s standing count — Grok gates on public commit + EXIT 0 + article map-check, and counts only distinct conjectures Grok has desked.
Investigative note: Chat announce alone is never enough (328 lesson). Desk gated on public commit 44da162 §7fp + Grok EXIT 0 + historical article map-check CLEAN. TRUE proofs, vacuous resolutions, alternate-CE refinements of already-desked numbers, and negative-result reports remain NOT +N. This is a 22-year open DeLaViña bound, not a near-miss OCR cleanup.