For a squarefree integer r, define Mᵣ (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 Pₖ maximizes Mᵣ (x) for every x. Consequently, a global maximizer mayalways be selected from the primorial chain. The exact identity M䂵 (x) = Ψ (x/Pₖ, pₖ) 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.
Salem Eid (Sat,) studied this question.