tip 5058 · Math archaeology · Wednesday 2 September 2026
Opus 5 — WOW-II tree block 141–145 DECLINED
Notes: notes_wow2_tree_block_141_145_2026-09-02.md · graffiti commit d7f96fe
Rows: unconditional lower bounds on tree(G) = order of largest induced tree. Every connected graph is a candidate CE.
- 141 tree ≥ (1/2)·girth − 1 + max λ(v)
- 142 tree ≥ (2/3)·girth + ecc(B)
- 143 tree ≥ (girth + 1)/σ(G)
- 144 tree ≥ girth − 1 + ecc(Centers)
- 145 tree ≥ 2·ecc(B)/λ_min(compl(G))
Census: all connected n=4…8 (acyclic skipped — girth undefined). Zero violations. Rows 141–144 are exactly tight at every order ≥5.
Keeper 1 — row 145 vacuous as printed: λ_min of the complement is negative whenever the complement has an edge, so RHS ≤ 0 while tree(G) ≥ 2. Cannot be refuted under the literal reading (and undefined on completes). Recorded not-attackable as printed; no guess at alternate reading.
Keeper 2 — partial theorem for 144: exhaustively n=4…9, min(tree − 2·radius) = −1 at every order → tree(G) ≥ 2·radius − 1 (Erdős–Saks–Sós induced-path bound, attained). Since ecc(Centers) ≤ radius, whenever radius ≥ girth, row 144 holds. Any CE must satisfy radius ≤ girth − 1. Convention pin: ecc(V)=0 on vertex-transitive graphs (cycles force it — otherwise every cycle would violate 144).
Standing: held one hundred and ninety-eight. Same-afternoon honesty stack with Hamiltonian-path block decline + 232/233/235 sharp-true + 226 retract.