Primorial Maximizers in Fixed-Radical Integer Families: Exact Reduction, Certified Adjacent Transitions, and a Saddle-Point Transition Framework

For a squarefree integer r, define M_r(x) = #{n ≤ x : rad(n) = r}. The fixed-support counting problem is classical. This paper studies the associatedextremal question: among all squarefree supports, which radical occurs most oftenbelow x? It is proved that, among radicals with exactly k distinct prime divisors, the kthprimorial P_k maximizes M_r(x) for every x. Consequently, a global maximizer mayalways be selected from the primorial chain. The exact identity MP_k(x) = Ψ(x/P_k, p_k) connects the problem with the counting function of friable integers. The paper reports internally certified exact transition data for the first elevenadjacent primorial pairs and develops a saddle-point framework for adjacentsmoothness transitions. Explicit first- and second-order logarithmic correctioncoefficients are derived algebraically. The higher-order uniform analytic expansionis presented conditionally, with its remaining contour-estimate obligations statedexplicitly. The literature position includes comparisons with fixed-prime-support counting,jumping champions, friable perturbation theory, and recent problems concerningequal integer radicals. The novelty claim is restricted to the fixed-radicalmaximization functional, its exact transition data, and its specific adjacent-smoothness transition model. Status: Revised preprint. The elementary reduction and exact identities are proved;the transition table is an internally certified computational result; the fullhigher-order analytic theorem remains conditional on the uniform estimates statedin the paper.