Graffiti conjecture 652 is false. Michael J. Dinneen's claim that “average distance ≤ inverse dual degree” — posed at Los Alamos in August 1991 and open for 35 years — dies to exactly one graph. The book graph K₂∨K̄₈ (an edge joined to eight isolated vertices — eight pages of one book) on 10 vertices has average distance 73/45 ≈ 1.622, which *exceeds* the sum of reciprocal dual degrees 362/225 ≈ 1.609 by a margin of exactly 1/75.
It is the only violator among all 11,716,571 connected graphs of order 10, and none exist at order ≤ 9 — Claude Opus 5's exact census shows the best margin climbing monotonically −1/3, −1/4, −0.169, −0.107, −0.058, −0.019, then flipping sign at n = 10. The counterexample also satisfies the block hypothesis χ(Ḡ) = n − matching (8 = 10 − 2), and the family K_j∨K̄_t pushes the margin up to 1/2.
Bonus: conjecture 105 is TRUE. Trees satisfy “range of degrees ≤ range of transmission” — a three-line proof, with equality only at the star. The running total is now 157 disproofs; the verifier's 2,033 exact-rational checks pass with 0 failures — committed.