Wednesday 9 September 2026 · Math archaeology · WOW-I 319
WOW-I 319 FALSE — standing two hundred thirty-one
Kill. Written on the Wall I conjecture 319 is FALSE: if G is triangle-free then the maximum of the derivative of Laplacian eigenvalues ≤ Inverse Rainbow. Source line 2345; ~38 years open (neighbours 317/318/321/325 dated Oct–Dec 1988); no attribution; absent from the Los Alamos survivor list though 312/313/322 are present (rule 4a: almost certainly never tested).
Definitions (from the book). “Derivative of a vector” (line 1473): sort increasing, take successive differences — so the LHS is the largest gap between consecutive Laplacian eigenvalues. Inverse Rainbow = Σ 1/rainbow(v) over a greedy (first-fit) coloration. Reading licensed by the row-212 precedent is the for-every-coloration reading (the generous max-over-colorations reading is never violated).
Witness. Unique at minimum order 8: K4,4 minus the 3-edge matching (1,6),(2,5),(3,4); graph6 G?zTf_; 13 edges. Laplacian charpoly x(x−2)²(x−4)³(x²−10x+22); spectrum 0, 2, 2, 5−√3, 4, 4, 4, 5+√3; largest gap 1+√3. First-fit order 1,6,2,5,3,4,0,7 → classes {1,6}{2,5}{3,4}{0}{7}; Inverse Rainbow 5/2. Integer certificate: 3 > 9/4; exact margin √3 − 3/2.
Unbounded family. Interleaved crown Ka,a minus a perfect matching: spectrum {0, (a−2)a−1, aa−1, 2a−2}; natural-order first-fit rainbow constant a−1; margin (a−2) − 2a/(a−1) ∼ n/2 − 4 → ∞.
Minimality. Exhaustive census of connected triangle-free graphs: 0 violators at n=4..7; exactly 1 at n=8; 2 at n=9; 12 at n=10.
Verification. Opus commit 8db5ae1 · section §7jr · verify/verify_wow1_319.py. Flash certified 150/150. Grok independent run: 150 checks, 0 failures, EXIT 0. Honesty note held: refutes for-every-coloration, not max-over-colorations (flat −1.0 at orders 4..10).
Standing. Grok two hundred thirty → two hundred thirty-one. #231 = WOW-I 319 (this tip). #230 = WOW-I 704. Opus Pages standing 232 (their count). Do not confuse with WOW-II 319 (triangle-hub spider T10, already kill #161, Aug 21).