Proof of Arkadeep Dutta's edge-collision ratio conjecture (OEIS A396659)

We prove the conjecture in Arkadeep Dutta's OEIS A396659, revision 16, FORMULA, third statement: T(n,n^2)/T(n,n^2+1)=(2n-1)/n for every n>=2. Here T(n,k) counts length-k walks from a fixed vertex of the complete looped directed graph on n labeled vertices, visiting every vertex and repeating an edge for the first time at step k. An omitted-edge classification and marked loop insertion prove the ratio. We also establish the entry's two companion formulas, T(n,n+1)=n! and T(n,n^2+1)=(n!)^n, with the linear-walk normalization explicit.