We introduce a binary-window framework for odd composite integers that survive wheel 30, that is, integers coprime to 2, 3, and 5. For such a composite integer n, we consider all contiguous binary windows in the binary expansion of n, reinterpret them as integers, and study the smallest window that is a multiple of a true factor of n. This defines a minimal multiple window Wmin (n) and induces a local/global dichotomy according to whether Wmin (n) = 1, p divides floor (n/2ᵏ) if and only if p (q mod 2ᵏ) < 2ᵏ. This gives a complete characterization of prefix divisibility by the smallest prime factor. To study the smallest prime factor specifically, we introduce the refined observable Wmin^ (p) (n), defined as the smallest binary window divisible by spf (n). In all tested cases up to 10⁶, Wmin^ (p) (n) existed for every composite survivor, and in all global semiprime cases the quotient Wmin^ (p) (n) /spf (n) coincided exactly with the cofactor. In particular, among the 61, 341 global semiprime cases found up to 10⁶, this law held with empirical accuracy 100%.
Ricardo Adonis Caraccioli Abrego (Tue,) studied this question.