Exceptional Sets for Semiprimes in Quadratic Intervals

For a prime k, let S_k(n) count the semiprimes kp (p prime) in the quadratic interval (n², (n+1)²), and let E_2(n) count the integers there with exactly two prime factors, counted with multiplicity. Companion papers formulated the conjectures that S_2(n), S_3(n) ≥ 1 and E_2(n) ≥ 1 for all large n, and verified them computationally to n = 10^7 and n ≈ 3.16×10^5 respectively; the pointwise statements are Legendre-class and remain open. This note proves the corresponding exceptional-set theorems. Under the Riemann Hypothesis, the number of n ≤ N with S_k(n) = 0 is O_k(log³ N), by the Selberg–Saffari–Vaughan variance bound. Unconditionally, it is O_{k,ε}(N^{53/130+ε}), where the exponent 53/130 = 0.4076... is computed from the Gafni–Tao exceptional-set bounds for the prime number theorem in short intervals, fed by the Guth–Maynard zero-density estimate and Heath-Brown's additive-energy bound A*(7/10) ≤ 235/39; thus quadratic persistence can fail only on a power-thin set. Finally, transferring the Matomäki–Teräväinen almost-all theorem for products of two primes by a sliding-window argument, we show that all but O(N / (log N)^δ) integers n ≤ N satisfy E_2(n) ≫ n / log n; in particular, almost every interval between consecutive squares contains an exact semiprime, with the conjectured density up to a factor log log n. We calibrate the three theorems against the exact computations of the companion papers.