Exact Semiprimes in Short Intervals: Prime Supply, Factor-Pair Multiplicity, and the Parity Barrier

We distinguish two related but fundamentally different short-interval problems: the occurrence of P_2-numbers, integers with at most two prime factors, and the occurrence of E_2-numbers, integers with exactly two prime factors. The former is accessible to classical weighted-sieve methods and is known in intervals much shorter than √x, while the latter is parity-sensitive in worst-case intervals. This paper proposes a heuristic framework for exact semiprimes in short intervals. The central observation is that local semiprime concentration is governed by two forces. The first is prime supply: after fixing a least prime factor p, exact semiprimes pq in (x, x+H] correspond to primes q in the scaled interval (x/p, (x+H)/p]. The second is factor-pair multiplicity: summing over possible least-prime-factor lanes contributes the harmonic multiplier sum_{p ≤ √x} 1/p ~ log log x. Together these forces predict

#{m ∈ (x, x+H] : Ω(m) = 2} ≈ H · (log log x) / log x.

However, this prediction is already vastly weaker than known almost-all results in typical intervals, and it does not by itself overcome the parity barrier in worst-case intervals. We therefore frame the main problem as a worst-case exact-semiprime gap problem: whether E_2-numbers have no √x-scale gaps, and in particular whether every sufficiently large interval (n², (n+1)²) contains an exact semiprime. An exact enumeration of Ω(n) for all n < 10^11 supports the framework: the local density matches the prediction up to a stable lower-order correction, the worst-case gap grows like (1/4)(log x)² — three to four orders of magnitude below √x — and every consecutive square interval below 10^11 contains an exact semiprime, with counts parity-symmetric and bounded well away from zero.