Maximum generic transitivity degree

Determine the exact value of τ(n), the maximum possible degree of generic transitivity of a connected definably primitive action on a set of Morley rank n, by proving τ(n) = n + 2.

Background

Generic multiple transitivity quantifies how highly transitive a definable group action can be on a set of given Morley rank, and it controls rank bounds for primitive groups via inequalities linking τ(n) and rk(G).

Examples from algebraic group actions (e.g., GL, AGL, PGL) suggest a tight upper bound of n+2, motivating the conjectured equality.