Imagine walking through a network with one rule: you cannot immediately go back along the edge you just used. The counting matrix for these walks need not behave like a symmetric matrix, even when the network itself is undirected. Its Jordan blocks describe chains of generalized eigenvectors, structure that a list of eigenvalues alone does not reveal. How large can those blocks become as the network grows? This anonymous, unrefereed candidate gives a constructive answer: the largest possible block size grows linearly with the number of vertices. There is a universal linear upper bound, and a family of simple connected graphs supplies a matching order of growth. The family can even keep every vertex degree below one fixed bound, three hundred and ninety-three thousand, two hundred and seventeen. This is not a claim about regular graphs, typical networks, or normalized random walks. The proof first constructs a small structured matrix pencil. It then replaces its algebraic entries by integer blocks and embeds the resulting action into a simple graph. The witness graphs are enormous and are specified by an edge-generating rule, not stored in full. Exact code checks finite pencils, small graphs, and deliberately broken examples. The universal conclusion rests on the written proof. The constants are not claimed optimal, and external specialist validation and historical priority remain unresolved. This is the Evidence Press release of fifth September twenty twenty-six. The paper and complete evidence package are linked on this page. This AI-generated OpenAI voice is a communication aid, not additional evidence.