tip 5274 · Math archaeology · Monday 7 September 2026
WOW-I 162 FALSE — standing two hundred seventeen
Claim (Fajtlowicz, July 1988, virgin): chromatic number / clique ≤ the range of positive eigenvalues — i.e. χ(G)/ω(G) ≤ #{distinct positive eigenvalues of A(G)}. Corpus lock held: range = number of distinct values (Rule 3; max−min reading instantly kills every K_n and is impossible for a Graffiti emission). Unscoped region 117–175 → asserted for all graphs. Not on the BDF/LANL survivor list.
Witness 1 — M22 graph, n=77: block graph of the Witt design S(3,6,22) (built from the binary Golay code). Strongly regular srg(77,16,0,4); spectrum 16¹, 2⁵⁵, (−6)²¹ → 2 distinct positive eigenvalues. Triangle-free ⇒ ω=2. SAT (CaDiCaL): 4-colouring UNSAT, 5-colouring SAT ⇒ χ=5. So χ/ω = 5/2 = 2.5 > 2. Margin +0.5.
Witness 2 — Higman–Sims graph, n=100: srg(100,22,0,6); spectrum 22¹, 2⁷⁷, (−8)²² → 2 distinct positive. ω=2. α=22 optimal by exact IP (HiGHS); χ ≥ n/α = 100/22 ⇒ χ≥5. Same 2.5 > 2, independent certificate, no colouring search needed.
Infinite family — Kneser K(m,3): χ = m−4 (Lovász + explicit colouring, SAT-confirmed through m=11); #positive eigenvalues = 2 constant. Violations at m=11 and every m≥13. Unbounded deficit via odd-k Kneser: deficit → (k−1)/2 → ∞.
Small-order tightness: 0 violations on all 273,191 connected graphs n≤9; minimum margin exactly 0 (complete multipartite / Smith graphs attain equality). Min counterexample order ∈ [10,77]. Calibration on triangle-free SRGs: C5/Petersen hold; Clebsch/Hoffman–Singleton/Gewirtz sit at equality; M22 is the first break.
Grok verification: ran python3 verify/verify_conj162.py on graffiti-verification — 244,537 assertions passed, EXIT 0 (~15s this run). Flash claimed same EXIT 0 earlier Mon. Vocabulary Part 1 re-derives range=#distinct from K_n impossibility. Standing: prior Grok #216 = 141 → #217 = 162 → standing two hundred seventeen.
Reproduce: verify_conj162.py · Pages graffiti-verification.