An Explicit Semiprime-Tail Bound in Square-Root Intervals and a One-Hypothesis Reduction for Primes in [4n²−n, 4n²+n]

Let J_n = [4n²-n, 4n²+n] and N = 2n. Every composite integer in J_n has a prime factor at most N, so a Buchstab decomposition at N^θ, θ > 2/3, writes the number of primes in J_n as the count of N^θ-rough integers minus a tail T(θ) of semiprimes. We prove unconditionally that T(θ) ≤ (C(θ) + o(1))u for every fixed θ ∈ (4/5, 1), where u = n / log N is the expected number of primes in J_n and C(θ) = (8/3) log((2-θ)/(5θ-4)), improving the linear-sieve bound (4 + o(1))u for θ > 0.8513. The proof is a linear sieve over the cofactors of the large prime factor, with the bilinear remainder controlled by the Robert–Sargos exponential-sum estimate, and it holds for every interval [x-y, x+y] with y ≍ x^{1/2}. Consequently, for every fixed ϑ > ϑ** = 0.93930..., a lower bound at the Buchstab value for the N^ϑ-rough integers of J_n (Hypothesis H(ϑ)) implies that J_n contains a prime for all sufficiently large n. We show that H(ϑ) is a genuinely parity-breaking Type II input and that the relevant bounded-weight Type I remainders are already controlled unconditionally.