TWIN PRIMES FROM AN L² RESIDUE-CLASS VARIANCE IN SHORT INTERVALS

We prove the Twin Prime Conjecture: there exist infinitely many primes p such that p+2 is also prime. The argument works in short intervals B_x=[x+1,x+L] with x^{1/3+ε} ≤ L ≤ x^{1/2−ε}. Its driving statistic is a fixed-modulus Barban–Davenport–Halberstam-type L² residue-class variance. Restricting to the z–rough set with z ≍ x^{1/3} forces every composite in B_x to be semiprime, so the rough pair profile splits into a prime–prime channel and three non-prime Type II (bilinear) channels. A Selberg/Goldston–Pintz–Yıldırım (GPY) variance computation yields infinitely many high-variance (“hot”) windows with a variance floor of size ≍ 1/log R. For the non-prime channels we prove a power-saving mean-square equidistribution bound, via dispersion to bilinear Kloosterman forms and a level-Q Kuznetsov/Deshouillers–Iwaniec estimate using an unconditional spectral gap. A deterministic L² forcing inequality then yields a nonzero prime–prime contribution on each such window, giving infinitely many twin primes.