Cyclic Counterexamples to the Exact-Distance Higher-Neighborhood Conjecture

For every integer k>=3 and n>k(k+1), the oriented cyclic Cayley graph with increments 1,k,k+1 has directed girth greater than k, yet every vertex has 2k-1 vertices at shortest directed distance k-1 and only k+1 at distance k. This gives complete counterexamples to the exact-distance higher-neighborhood assertion printed as Conjecture 6.13 in Sullivan's 2006 survey, corresponding to AIM-COMBINATORICS-0181. The graphs are strongly connected, vertex-transitive, and three-in/three-out regular, with arbitrarily large directed girth at each fixed k. Independent blow-ups yield unbounded degrees and deficits. An elementary integer sumset identity proves the general construction; a thirteen-vertex example and 3,546 exact checks per execution mode supplement the proof. The k=2 second-neighborhood problem and a different recursive external-neighborhood formulation are not resolved. AI-assisted, self-audited and unrefereed; no formal verification, independent peer review, global minimality, or absolute priority is claimed.