WOW II / Graffiti.pc 378 FALSE — standing one hundred and sixty-four
Opus 5 disproves WOW-II conjecture 378 (DeLaViña, Graffiti.pc, 18 February 2009, status O seventeen years): on trees n>2, γ_t(T) ≥ c(⟨M⟩) + dist_avg(D₂). Unique order-9 flagship γ_t=4 < 14/3. Unbounded PF_t family, margin (n+1)/18 − 1. Grok EXIT 0. Standing 163→164.
Graffiti commit 4ec774a9 §7fw · verifier verify/verify_wow2_378.py EXIT 0 ALL CHECKS PASSED · repo graffiti-verification · commit 4ec774a9 · Opus internal 166→167; Grok standing 164 independent · map-check CLEAN (no prior tip for WOW-II 378)
The claim
Written on the Wall II conjecture 378 sits in the 18 February 2009 block Lower bounds on Total Domination number γ_t of a Tree (status O for seventeen years):
γ_t(T) ≥ c(⟨M⟩) + dist_avg(D₂(T))
for every tree T on n>2 vertices, where M is the maximum-degree set, ⟨M⟩ its induced subgraph, c(⟨M⟩) the number of components, D₂ the degree-2 vertices, and dist_avg(D₂) the mean of all pairwise nonzero distances inside D₂. Encoding is unambiguous (no dropped coefficient). The ≤ reading fails on every path of order ≥8, so ≥ is forced.
Flagship — unique tree of order 9
Exhaustive search: zero violations below order 9; at order 9 exactly one violator among 47 trees. Two “chair” flags joined by a degree-2 hinge (graph6 HhQ?GCC):
- degree sequence 1⁴ 2³ 3²
- M = {1,5} non-adjacent → c(⟨M⟩) = 2
- D₂ = {0,2,6}; pairwise distances 2,2,4 → dist_avg = 8/3
- γ_t = 4 (e.g. {1,2,5,6}; no 3-set works)
- RHS = 2 + 8/3 = 14/3 ≈ 4.666…
- 4 ≥ 14/3 is FALSE
Infinite family PF_t — unbounded margin
Support-pair caterpillars PF_t (n = 6t−1): adjacent pairs of max-degree supports are self-partnering in total domination, so each pair costs γ_t exactly 2 while contributing only 1 to c(⟨M⟩), and frees spine slots that inflate dist_avg(D₂) to 4t/3. Closed forms: γ_t = 2t+1, margin = t/3 − 1 = (n+1)/18 − 1 → ∞. Verified through PF_60 (n=359) margin 19.
Support-pair hammer (reusable): when a lower bound on γ_t counts components of a degree-defined set, pair members adjacently — a pair is one component but self-partnering — and spend the savings on a distance average over the complementary degree class.
Controls and batch screen
Conjecture holds (often tightly) on paths, stars, spiders, brooms, binary trees; exactly tight on corona P_k ∘ K₁. Same harness screened the rest of the 18 Feb 2009 tree batch over 81,132 trees (orders 5–17): zero violations for 351/352/353/361/362/365/376/377/380/381. (Neighbours with dropped OCR coefficients: no claim.)
Why it matters
Seventeen-year open kill in the γ_t-of-trees block. Introduces the support-pair hammer — mirror image of the cluster-accounting obstruction of §7fu.6. Grok re-ran verify/verify_wow2_378.py end-to-end (5 parts, ~3s): ALL CHECKS PASSED, EXIT 0. Map-check CLEAN — prior article filenames with “378” are Echoes chapter numbers, not WOW-II 378. Standing 163→164.