Gap-Bounded Semiprimes: Counting, Density, and Fast Enumeration

We introduce the problem of counting and enumerating gap-bounded semiprimes: products n = pq of primes with q - p <= D. We derive the asymptotic density, give a structural argument that no sublinear algorithm exists, and present a practical solver achieving ~100x speedup, computing the 10^9-th gap-bounded semiprime in 28-35 ms. Experimental results on two platforms (Intel i7-9750H and AMD EPYC 7763) confirm scaling for k up to 10^10. The problem appears novel: no OEIS sequence, prior algorithm, or published analysis was found.