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 counting primes in J_n is a pure sieve problem at the natural boundary. Write u := n / log N, the prime number theorem's prediction for the number of primes in J_n. A Buchstab decomposition at N^θ splits this count into a rough-number main term and a semiprime tail T(θ); we prove, unconditionally, that the tail is of the correct order of magnitude, T(θ) ≤ (C(θ) + o(1)) u, with the explicit elementary constant C(θ) = (8/3) log((2-θ)/(5θ-4)), for every fixed θ ∈ (4/5, 1). The proof runs a linear sieve over the cofactors of the large prime, with the bilinear remainder controlled by the three-dimensional exponential-sum estimate of Robert and Sargos after a Vaughan decomposition of the prime variable; the certified sieve level N^{(5a-4)/4} against a prime block N^{2-a} is optimal for the method. Because the tail is now a theorem rather than a hypothesis, counting primes in J_n reduces to a single medium-range hypothesis: a Buchstab-asymptotic lower bound for the N^ϑ-rough integers in J_n. This reduction succeeds for every fixed ϑ > ϑ* = 0.93930..., the root of an explicit equation; granting the hypothesis at any such ϑ, the interval J_n contains a prime for all sufficiently large n. We state the variance-strength equidistribution estimate for the square-centered residue classes that underlies this hypothesis, prove that it controls all sieve remainders at level N^ϑ, and delimit precisely the combinatorial step that remains open.