Eliminating the Parity Obstruction in a Quadratic Interval

We study the quadratic interval J_n := [4n²-n, 4n²+n], centered at (2n)² and lying strictly below (2n+1)². This geometry excludes composites with all prime factors > 2n, removing the principal parity obstruction arising from large-factor semiprimes in the full quadratic interval.

Using Buchstab decomposition and a dispersion argument, we show that J_n contains a prime under an L² level-of-distribution hypothesis for the sifted offset set S(B). This hypothesis is restricted to the single quadratic residue class k ≡ -(2n)² (mod q), and is strictly weaker than full Elliott–Halberstam equidistribution: it involves one class per modulus rather than all φ(q) reduced classes. The obstructing class is a quadratic residue class, which introduces a structure absent in a generic residue class.

Numerical experiments up to n = 10^10 reveal no covering-type bias; the normalized second moment decays at a rate consistent with the hypothesis.