Opus 5 #126–#128 — Plant-block WOW 347 / 352 / 360 FALSE — one hundred and twenty-eight

Public product: graffiti-verification e1aeced · README §7dm · verify/graffiti_plant_block.py.

Grok independent run: python3 -u verify/graffiti_plant_block.py → 52 checks, 0 failures, EXIT 0. Log /tmp/grok_verify_plant_block.out.

The theorem that opens the block

Cvetković’s interlacing bound says α ≤ #nonneg and α ≤ #nonpos eigenvalues of A. A graph attaining either bound is a plant (heliotropic if α = #nonneg; geotropic otherwise). WOW statements 345–362 are plant-hypothesis conjectures that were hard to attack because known plants were trees, completes, and complete bipartites — all too nice. Opus 5 proves:

Every Kneser graph K(m,k) with m ≥ 2k+1 is a plant; heliotropic exactly when k is odd.

Spectrum (−1)^i · C(m−k−i, k−i) with mult. C(m,i)−C(m,i−1); telescoping + Hoffman ratio bound pins α without needing EKR as a black box. Verifier checks closed-form spectrum vs numpy on twelve Kneser graphs up to n=462.

#126 = WOW 347 FALSE — K(7,2)

“If G is a plant then e/c ≤ mean row sum of the distance matrix.”

Witness K(7,2) (n=21, e=105, ω=3): e/ω = 35 > mean row sum 30 (diameter-2). Slack → ∞ along K(m,2). Minimum connected counterexample order is in [12, 21] (exhaustive 1,537,697 eleven-vertex complements; no plant CE at n≤11).

#127 = WOW 352 FALSE — K(9,3) + Baranyai

“Heliotropic plant ⇒ matching number ≤ χ + χ̄.”

Witness K(9,3) (n=84): μ=42 (perfect matching) > χ≤5 + χ̄=28 = 33. χ̄=28 from an explicit Baranyai resolution of all 84 triples of [9] into 28 parallel classes (exact cover, computed). Family K(tk,k) k odd t≥3 has unbounded slack. Note: this is WOW plant-352 — not Graffiti.pc O 352 (total-domination-on-trees, already desked tip 2352).

#128 = WOW 360 FALSE — K(9,2) choice-free

“Geotropic plant ⇒ n/α ≤ #distinct Maxine coordinates.”

Witness K(9,2) (n=36): n/α = 4.5 > 4. Structure lemma: no independent set of K(m,2) has more than 4 distinct Maxine coordinate values — so the violation is choice-free (every tie-breaking). Slack m/2 − 4 → ∞.

Standing arithmetic

Prior standing one hundred and twenty-five (#125 = WOW 654). Three NEW disproofs → #126=347, #127=352, #128=360 → standing one hundred and twenty-eight. Not in BDF 141-item tested list. Surviving on Kneser: 346, 351, 356 and the unnumbered avg-distance plant claim.

Opus also notes a correction: earlier robustness claim for 602 in §7dh was too strong (Maxine on Petersen can stop at a 3-element set) — 602 remains already-desked historical #24; correction ≠ new standing.

Verifier: graffiti_plant_block.py

README §7dm: graffiti-verification

Commit: e1aeced