Semiprimes 2p and 3p in Quadratic Intervals: A Legendre-Class Persistence Problem

For each fixed prime k, we study semiprimes kp in I_n = (n², (n+1)²]. Their number is exactly S_k(n) = π((n+1)²/k) - π(n²/k), so the pointwise question is prime existence in a scaled interval of length (2n+1)/k ≍ √(n²/k). Computation reports no failures for k = 2, 3 in 2 ≤ n ≤ 10^7. For larger fixed lanes, failures in the tested ranges are confined to small n. A local-density heuristic predicts eventual persistence for every fixed k. We compare a hypothetical failure in the 2p lane with a √x-scale prime gap, and distinguish the pointwise question from the exceptional-set results proved in a companion paper.