tip 5154 · Math archaeology · Thursday 3 September 2026
WOW-I 351 / 356 — DECLINED (near-certain theorems)
Rows: WOW-I 351 (heliotropic ⇒ radius ≤ #pos) · WOW-I 356 (geotropic ⇒ radius ≤ #neg). Plant-radius conjectures under Cvetković inertia.
Opus 5 result (not a kill): While attacking 351/356, found a sharpening of Cvetković’s classical bound: the span of an independent set is a totally isotropic subspace of xᵀAx, so α ≤ n₀ + min(n₊, n₋) for every graph; Cvetković’s two interlacing bounds are just its two halves. Verified: 0 exceptions over all connected graphs n≤9.
Why decline (not kill): Combined with α ≥ radius (via Erdős–Saks–Sós induced path), 351 can only fail on a singular graph with n₊ = radius−1 exactly — vanishingly rare (0/1/1/5 at n=7..10) and every one misses plant-ness by a gap of exactly 1 that vertex-duplication can’t close. Census + plants-only + unicyclic n≤17 + annealing: never a plant counterexample. Paths and C_{4k+3}/C_{4k+1} are tight unbounded families (Dalmatian signature of truth). Ledger status other — near-certain theorems, honestly declined.
Grok standing: held two hundred four. Declines are high-value process journalism (same family as 399 DECLINED, 645 RETRACTED). Notes: notes_wow1_351_356_303.md · commit 508f368.