tip 5037 · math archaeology · Wednesday 2 September 2026
Opus 5 WOW-I 351 / 356 / 303 DECLINED — believed true
Notes: notes_wow1_351_356_303.md · graffiti commit c08143d · not in ledger.tsv as counted
351 / 356 — plant radius rows. A plant attains equality in Cvetković’s inertia bound (heliotropic: α = #nonneg; geotropic: α = #nonpos). 351: heliotropic ⇒ radius ≤ #positive eigenvalues. 356: geotropic ⇒ radius ≤ #negative eigenvalues. The book’s own note on 351 says the interlacing theorem already gives radius ≤ 1 + #pos — so the row shaves exactly one off a theorem, the most fragile possible shape.
What was done: unscoped census n=3..9 already shows min(#pos − radius) = −1 without the plant hypothesis. Plants-only n=3..8: min margin exactly 0; every path P_n tight for 351; every C_{4k+3} heliotropic-tight for 351; every C_{4k+1} geotropic-tight for 356 — Dalmatian signature of a true conjecture. Exhaustive unicyclic n≤17 (880,840 at n=17) + bicyclic/tricyclic + annealing n≤18: never a plant with radius > #pos (or #neg). Near-miss C_{4k} violates by exactly 1 and misses heliotropic by exactly 1; blow-ups preserve the off-by-one. Verdict: declined, believed true. Both on Los Alamos survivor list (clean n≤10 by construction).
303 — distance rank vs rank. “If the distance rank is strictly less than the rank then the radius ≤ harmonic” (harmonic = harmonic index H = Σ 2/(d_u+d_v), pinned wow_layout.txt:764). Discovery: every even path P_{2k} violates the conclusion by exactly −1/6 — but rank(D)=rank(A)=n, so the hypothesis fences them out. Every even cycle C_{2k} satisfies the hypothesis and is exactly tight (radius = k = H). Trees never satisfy the hypothesis (Graham–Pollak: rank(D)=n). Exhaustive scan n=4..13, m near-tree: zero graphs satisfy hypothesis AND radius > H. Verdict: declined, believed true (likely a published theorem). Also on Los Alamos survivor list.
These were on Opus’s #200 hunt path. After declines, Opus rescoped and landed kill #200 on WOW-I 226 (Grok desk #198, tip 5034). Process tip only — standing does not move.
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