We study integer families with fixed radical and the transition between adjacent primorial supports. The counting problem is reduced exactly to friable-number counts, and an exact renewal identity decomposes the contribution of the newly introduced prime according to its exponent. Using published local ratio estimates for friable integers, we prove that every adjacent primorial pair admits a reference one-count crossing and that, along any selection of such crossings, the normalized exponent profile converges unconditionally in total variation to the geometric distribution π (a) =2−a, a≥1. (a) =2^-a, a1. π (a) =2−a, a≥1. Thus, the first-order geometric law does not require a higher-order saddle-point or Edgeworth hypothesis. We also develop a closed Hildebrand–Tenenbaum saddle-point model, exact renewal and log-concavity identities, affine-cancellation estimates, Green-kernel and martingale representations, and reproducible exact computations through the transitions k=13k=13k=13 and k=14k=14k=14. The reported transition counts are independently reproduced using two distinct meet-in-the-middle decompositions. A second-order analysis predicts dTV (πk, 2−a) ∼38k. dₓₕ (ₖ, 2^-a) 38k. dTV (πk, 2−a) ∼8k3. This rate is stated conditionally upon a uniform one-term Edgeworth expansion for friable counts on a logarithmic dilation ladder. The paper explicitly separates unconditional results, exact computations, conditional refinements, and remaining open problems.
Salem Eid (Sun,) studied this question.