A Six-Vertex Counterexample to Vertex-Averaged Ordered Reachability

We give two loopless directed graph layers on six vertices, each with constant outdegree two, having no rainbow directed cycle and hence no increasing rainbow cycle. For the fixed label order 1<2, the sizes of the sets reachable by possibly trivial increasing paths are 5,5,5,5,5,4. Their average is 29/6<5=1+delta_1+delta_2. Thus the uniform-start-vertex interpretation of the ordered-reachability conjecture attributed to DeVos in Sullivan's 2006 survey is false. Uniform blow-ups give examples on 6m vertices with average 1+23m/6, below the proposed bound 1+4m for every m>=1. The proof is an explicit neighborhood calculation and does not depend on the numerical search that located the example. Unrefereed preprint. The original source does not specify its averaging variable. This result addresses a fixed label order and a uniform average over starting vertices, not averaging over label orders. It does not disprove the Caccetta-Haggkvist conjecture or its unordered rainbow-path analogue. The source package includes an exact standard-library Python verifier and archived results. No independent peer review, formal verification, or absolute priority certificate is claimed. AI-assisted tools supported research, computation, proof development and writing; the author is responsible for the final text. Corpus identifier: AIM-COMBINATORICS-0177.